According to the meaning, the French idiom would be:

Il ne faut pas vendre la peau de l’ours avant de l’avoir tué.

This comes from L’Ours et les deux Compagnons, which is one Fable by Jean de la Fontaine, and ends with:

Il m’a dit qu’il ne faut jamais.

Vendre la peau de l’Ours qu’on ne l’ait mis par terre.

L’expression est trop moderne pour avoir un équivalent exact. Je dirais bien :

Tant qu’il y a de la vie, il y a de l’espoir.

(As long as there is life, there is hope.) Ce n’est pas un équivalent parfait, mais je le trouve déjà mieux que « il ne faut pas vendre la peau de l’ours avant de l’avoir tué » (qui a un équivalent exact : don’t count your chickens until they are hatched).

Si l’expression est utilisée à propos d’une situation en cours, on peut aussi dire tout simplement

Tout n’est pas encore joué.

ce qui traduit très bien le sens, mais n’est pas tout à fait au même niveau de langue (l’expression française est un poil plus formelle).

For some "colorful" examples:

Il ne faut pas compter les œufs dans le cul de la poule.

C’est à la fin de la foire qu’on compte les bouses.

« Qui vivra, verra » traduit la même idée.

The “heroine” fat lady is a stereotype of valkyrie Brünnhilde in the R. Wagner opera Götterdämmerung(Damnation of the Gods), the last opus of the Ring works. The world ends thereafter. Considering the setting, maybe something along the lines of:

Tant que le rideau n’est pas tombé(sur
scène ou sur le monde), la pièce n’est pas terminée.

In so many words “still on stage”.

For what it’s worth at this late date, I agree with @Gilles’ comment under the question that “It’s not over until the fat lady sings” (hereinafter “the F.L. Expression”), and it’s shortened version as mentioned by Benjol: “it [ain’t] over ‘til it’s over,” are both neutral expressions (“ neutre vis-à-vis du caractère bénéfique ou non de la situation”) that are merely stating the obvious that outcomes can change, for better OR for worse, right up until the outcome becomes officially final.

The F.L. Expression is often used by coaches of sports teams (and politicians) to either:

1) CAUTION their team (or campaign workers) not to prematurely celebrate victory just because they are ahead in the game (or in the pre-election polls) because the game (or election) is not yet “over” or finished and there is still time to mess-up and lose the game (or election);

OR

2) ENCOURAGE their team not to prematurely give up/admit defeat just because they are losing because there is still time to come from behind and win.

Cautioning against “counting chickens” or “selling bear fur” prematurely would certainly capture the F. L. Expression’s meaning in #1, where it means “cautioning against premature victory celebrations”, but I’m not convinced that they are neutral enough to accurately capture the F. L. Expression’s meaning #2, where it means “encouragement not to prematurely admit defeat.”

(Of course, if “don’t sell the bear’s fur ‘til it’s dead” also means “don’t mourn yet for the poor bear because there is still time to save him,” then it might also capture meaning #2 of the F. L. Expression. However, my French wife assures me (and right or wrong, I usually listen to her) that the “bear fur” expression is not neutral, i.e., that it is ONLY a caution against acting foolishly and celebrating prematurely and that it is not meant to be interpreted to include the notion of “it’s not too late to save the poor bear.”)

The popularization (although not its origination) of the F. L. Expression has been attributed to Dan Cook, a TV sports editor, and its shorter version to Yogi Berra, a well-known baseball player, coach, and coiner of expressions.

In light of the the above-mentioned indirect connections to the world of sports and its direct connection with theatre/opera, I think that suitable French translations of the neutral notion of the F. L. Expression could/should also connect with either sports/games(competitions) or else the world of theatre, which, in my opinion, Gilles’s « Tout n’est pas encore joué » and Amphiteóth’s « Tant que le rideau n’est pas tombé, la pièce n’est pas terminée » both do very nicely.

Additional neutral, sports-related options could include the following translations/paraphrases of something that Rocky Balboa said in “Rocky V (“a fight ain’t over till you heard the bell”):

« C’est pas fini tant que la cloche n’a pas sonné ! »

or

« C’est pas fini tant que le gong n’a pas retenti. »

And finally, in the context of the “sport” of romance, I think that Hélène Ségara captures nicely the neutral notion near the end of her song “Avant la fin”:

Avant la fin, [entre nous deux,] rien n’est fini…

In a strictly cautious meaning :
“Il ne faut pas chanter [or crier] victoire trop tôt”.

For a neutral meaning I would translate quite freely : “Tant que ce n’est pas fini, ça continue”

La bonne expression est celle-ci:
"C’est à la fin de la foire qu’on compte les bouses"