Does the battery specifically say SERVICE battery? Can you post a picture of the battery capture with something like the coconut battery program? The reason I ask is that I'd like to see the condition of the battery.
Don't feel bad about the applecare running out. Once it reaches around 900 cycles, it is really hit or miss if apple will replace it. Those who have had their battery replaced by apple tend to have good experiences but you want to BACK UP EVERYTHING before you send it away.
One other consideration - the computer is now over 3 years old and if you reached that many cycles then it has probably had a TON of use. Consumer Reports and other publications recommend that after 3 years of use, it is more cost-efficient to replace a laptop instead of repairing it. While the battery can be fixed, more issues may arise and many issues cost a great deal to repair.
My advice as an Apple Cert tech (2011 10.6 certification, old but relevant
) would be to look into replacing the macbook. You could sell it for probably $400-$500 even with the battery issue.