Can I install an iPhone 5 Battery in an iPhone 5S?

I was just given an old iPhone 5s that is splitting at the seems. Clearly it has a battery that is going bad. But the phone still power up and works.

Unrelated to that, I have an old, unused iPhone 5 battery sitting on my desk for like a year now. I bought it for a different phone but that phone died before I got around to installing it.

Can I put the 5 battery in the 5S?

Looking at photos, they look physically identical.

Looking at specs, it looks like the 5 battery has a bit less capacity, but thats no big deal.

What do you think, will it work? I wouldn't be using it as a phone, probably just as a clock or something. Maybe a remote. I can get SOME use out of it I'm sure.