Modify li-ion charger to charge iPhone battery?

Disclaimer: this question might show a complete lack of understanding of the finesse of electronics.

My iPhone 4S' charging circuit is not working properly, so I am looking at convenient ways of charging the battery outside of the phone.

I've found a circuit diagram posted here by oldturkey03, and I will give that a shot.

Then I had another thought; I do have old li-ion chargers, for digital cameras and such. If one of these has the correct voltage (3.7V) and the correct current rating (<=1000mA), would it be up for the task, or could this still break my battery?

I've been using this setup many times now and it's working great:

I have very simply connected the + and - terminals of the charger to the corresponding pins of the battery. Most of the jumper cables are used to keep the battery in place.

By a strike of luck, the small jumper wires fit very well into the battery plug.