My translation--one that slightly modifies the original-- is Vincere est pervincere detrimentos. I think however your question will generate a lot of comment.
You should realize that Latin does not have the rich vocabulary of English; as such the common word vincere - "conquer" also includes the idea of "succeed". In general then, "overcome" and "succeed" use the same word in Latin.
It's a matter of opinion then if the same word can be used in both locations of the maxim. I'm generally against this, but classical Latin offers only a few alternatives to vincere. I ultimately settled on the related pervincere - "rout, utterly conquer", since I think the maxim's idea is that the defeat of adversity is essential, and so is of a greater degree than the resultant success. Evincere is also a possibility here, but my taste is that the prefix per is more emphatic.
I'm not very happy with detrimentum - "adversity". One should note first off that for the Romans many nouns we consider abstract were more concrete. Glibly, a Roman wouldn't say he overcame adversity, but that he overcame adverse things. Thus, I choose the plural detrimentos. However there are more specific alternates like calamitates, the wordy res arduas or fortunam malam - "bad luck".
In summary, the maxim presents a challenge in classical Latin becasue abstract ideas like the nuance between "suceed" and "overcome" and the notion of "adversity" need to be converted into concrete equivalents. Late/Medieval Latin is better equipped with abstract terms, and I'm interested in the board's opinions.