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Both the spellings are correct; which one is used depends on the context, and the English dialect.

As reported by the New Oxford American Dictionary:

In British English, the normal spelling in general contexts is judgement. However, the spelling judgment is conventional in legal contexts, and standard in North American English.

Current preferences

To judge from the Ngram chart for judgement (blue line) versus judgment (red line) for the period 1675–2008, judgment has been the more common spelling for more than 300 years:

The preference exists (although with a steadily narrowing gap between 1875 and 1990) for sources that Google Books classifies as "British English":

and for sources that Google Books classifies as "American English":

So it would be inaccurate to assert that judgement is the standard British English spelling, although that spelling clearly is much more common in British English than in American English.

Background of the British preference

It bears noting that Samuel Johnson, A Dictionary of the English Language (1756) lists only the spelling judgment:

JUDGMENT. s. {jugement, French} 1. The power of discerning the relations between one term or one proposition and another. Locke. 2. Doom; the right or power of passing judgment. Shakespeare. 3. The act of exercising judicature. Addison. 4. Determination; decision. Burnet. 5. The quality of distinguishing propriety and impropriety. Dennis. 6. Opinion ; notion. Shakespeare. 7. Sentence against a criminal. Milton. 8. Condemnation. Tillotson. 9. Punishment inflicted by providence. Addison. 10. Distribution of Justice. Arbuthnot. 11. Judiciary laws; statutes. Deut[e]r[onomy]. 12. The last doom. Shakespeare.

Evidently, Johnson considered that the spelling judgment applied to all of these meaning, with no splitting between legal senses of the word and senses involving personal qualities of insight or discrimination.

The third edition (1867), the fifth edition (1773), and the tenth edition (1792) of Johnson's Dictionary retain the same entry and the same spelling.

The change in British preferences appears to have been prompted by the Rev. J. H. Todd's revision of Johnson's Dictionary (1827), which retains Johnson's definitions but changes the spelling to judgement.

Joseph Worcester, A Dictionary of the English Language, volume 1 (1860), page 794, has this interesting note on the orthography of the word:

The following words, abridgment, acknowledgment, and judgment, are to be found, with the orthography here given, in the English dictionaries which preceded the publication of Mr. Todd's improved editions of Dr. Johnson's Dictionary. Todd altered Johnson's orthography of these words by the insertion of an e, thus, abridgement, acknowledgement, judgement; and he remarks, "Several authors have revived this orthography, retaining the e to soften, as Lowth observes on judgement, the preceding g, and as Johnson himself analogically writes lodgement."

The English dictionaries of Jameson and Smart, which have appeared since the publication of Todd's edition of Johnson, also retain the e; and Smart remarks, in relation to the three words in question, that "Todd restores the e in order that they may not exhibit the otherwise unexampled irregularity of g soft before a consonant;" and he "adopts the more correct, however less usual, spelling;" and in his Grammar he says, "It is certainly better to write judgement, abridgement, acknowledgement, &c., than judgment, &c., since, by the general laws of pronunciation, g is hard in terminating a syllable." Many respectable writers now insert the e in these words. The omission of it, however, has been hitherto, and continues to be, the prevailing usage; but it is perhaps not very improbable that the more consistent orthography may yet be generally adopted.

Background of the American preference

Meanwhile, the U.S. spelling preference went in the opposite direction. Noah Webster's first dictionary, (1806) has this entry for judgement:

Judgement, n., a sentence, decision, opinion, skill

Webster doesn't present a rationale for this break with Johnson's orthography, and his spelling here is inconsistent with his treatment in the same dictionary of acknowledgment and abridgment.

But in his voluminous An American Dictionary of the English Language (1828), Webster (again without explanation) reverses his judgement to judgment:

JUDGMENT, n., {Fr. jugement} [1.] The act of judging ; the act or process of the mind in comparing its ideas, to find their agreement or disagreement, and to ascertain truth ; or the process of examining facts and arguments, to ascertain propriety and justice; or the process of examining the the relations between one proposition and another. Locke. Encyc. Johnson. ...

... and so on through seventeen more definitions, many of them extracted from senses of the word that Webster finds in Biblical quotations.

Although Webster doesn't explain the flip-flop, he appears elsewhere to be quite critical of what might be termed the superfluous silent e (as in axe, ermine, iodine, jasmine, and medicine, all of which words Webster at one time or another rendered without the final e). The real surprise is that he broke with Johnson's spelling of judgment back in 1806.

Conclusion

Insofar as British English may be said to favor judgement and American English judgment, responsibility for those preferences seems to rest with the prescriptive orthographical choices of J. H. Todd and Noah Webster.

The judgment/judgement orthographic pair is especially interesting in that the dictionary preferences in British English and American English reversed between 1800 and now, with many British dictionaries shifting from Johnson's judgment to judgement and virtually all U.S. dictionaries shifting from Webster's judgement to judgment. The only other clear instance of this type of double reversal in British and U.S. English spellings that I'm aware of is waggon/wagon—although a case could be made for sceptic/skeptic, too.

I grew up spelling it with the "e" in the middle. I can almost remember when this changed, but not quite. Leaving out the "e" defies the rules of pronunciation I learned. Without the "e" the "g" in "judgment" becomes the hard "g." I believe the only correct spelling is "judgement." I don't know who changed it, or why; it just needs to be changed back.

From Merriam-Webster:

Judgment can also be spelled "judgement," and usage experts have long disagreed over which spelling is the preferred one. Henry Fowler asserted, "The OED [Oxford English Dictionary] prefers the older and more reasonable spelling. 'Judgement' is therefore here recommended." William Safire held an opposite opinion, writing, "My judgment is that Fowler is not to be followed." "Judgement" is in fact the older spelling, but it dropped from favor and for centuries "judgment" was the only spelling to appear in dictionaries. That changed when the OED (Fowler's source) was published showing "judgement" as an equal variant. Today, "judgment" is more popular in the U.S., whereas both spellings make a good showing in Britain.
Source: https://www.merriam-webster.com/dictionary/judgement#note-1

From "Writing Explained":

When Noah Webster was drafting the first American English dictionary, he sought to simplify many words from the traditional British English. Webster is usually credited with the creation of many American spellings that have fewer letters than their British counterparts. Abridgement/abridgment and acknowledgement/acknowledgment are a few examples of this, as are color/colour and flavor/flavour.
The word judgment, however, has been around just as long as the lengthier judgement and was in use long before Webster wrote his first dictionary, so while Webster didn’t invent the shorter judgment, he can still be credited with popularizing it in North America.
Most American dictionaries and usage guides prefer the shorter judgment. The AP Stylebook, The American Heritage Dictionary, and Merriam-Webster’s Dictionary all prefer judgment.
Interestingly enough, despite the fact that judgment is seen as a primarily American spelling, it is still the preferred form for legal works in British English.
Source: https://writingexplained.org/judgment-or-judgement-difference