I had this done just last week. Be aware that it does take some time (about a month) to get the replacement.
This is what happened for me: In early Feb, I called Apple Support to ask about the replacement battery. The rep asked me a bunch of questions about behavior, usage, what iOS I was on, etc. Based on my answers, he agreed that it was a battery issue and submitted an order for a new battery and said my local Apple store would call/email when the battery came in, usually 2 weeks for the battery to arrive at the store. From that point I would have 7 days to respond or they would give the battery to someone else.
Fast forward 3+ weeks to late Feb, the Apple store calls and I brought my phone in for the replacement. Arrived at 7pm, left the phone with them while I walked around the mall. They told me to return at 8:40. The phone was ready at 9:15pm. Total cost $29. There were lots of people in the store waiting for battery replacements and it was a little chaotic. YMMV, maybe the store I went to was just pretty busy overall. But life has been pretty good so far with the new battery!
Good luck!