If you are willing to brave the tiny screws that come with opening up the apple watch, you could attempt a DIY battery swap.But don't be surprised if you accidentally break something - apple watch repairs are notoriously difficult, especially the steps for getting the screen off and back on without cracking it.
When my series 0 battery completely died in 2018, Apple Support said they would charge $250 for out of warranty repair (I think because they said it wasn't only the battery that had to be replaced). Obviously, that is much more than the watch is worth so I just got a new one. It looks like from their Watch Service page that it should cost $79 for out of warranty 'battery service fee'.
Of course, it wouldn't hurt to check with Apple Support yourself, and there might be a free repair program of some kind, but I wouldn't count on it.
[Link Edited by Moderator]