Is there a word for a promise so solemn that the only way you would fail to keep it is if death stopped you?

A vow.

A vow is a solemn promise.

Oath is another word, but it usually suggests that the promiser swears on something holy to him/herself. So for some that would express a very deep commitment to the promise.

Unbreakable Vow

Cross-my-heart-and-hope-to-die? Blood oath?

In DrBill’s dictionary, “I give you my word” is a take it to the grave commitment.

I use the term “sacred promise”, meaning “by all I hold sacred”. “Oath” has legalistic overtones; “Vow” likewise religious. To me, the word “oath” would have the meaning you are looking for, but it doesn’t have the same weight to the civilian mind.

@Rarebear Unbreakable Vow works for Harry Potter fans. As @marinelife and @Seaofclouds note, vow comes as close as any single English word I am aware of. I guess there isn’t a word for a promise so binding that death or release from it by the other party to it is the only way out. If there were, way over half the current population of Earth would be dead. :-).