The instructions I followed are available here.
I bought a battery and matching toolkit (two small screwdrivers, plastic pry-tool and two tiny Phillips-head screws to replace Apple's annoying pentalobe screws,) from iFixit for $30. It took about 10 minutes (not counting the time to sync and backup the phone) to complete this repair. Far better than sending it to Apple, who would charge $80 and hold on to the phone for a week.
All of our battery problems appear to have gone away, thanks to this relatively simple fix.