You need to be very careful on this step, if you touch the battery with the soldering iron the battery may explode.

Do NOT heat up the battery too much or it may also explode, you need to be quick doing the soldering.

Remove the clear tape from the old battery.

De-solder the old battery circuit from the old battery.

You will need this circuit, so be careful not to damage it!

De-solder the battery circuit that comes with the new cell, and solder the battery circuit from the old ring battery to the new battery. This will convert your new cell to a cell compatible with the Ring's charging circuitry.

Make sure to observe polarities. The solder pad with the red wire next to it goes to the '+' poll on the battery, and the solder pad with the black wire goes to the '-' poll on the battery.