I'm not sure what you're talking about when you say "the whole thing". The removable battery is here: MacBook Battery
Here's how it's replaced: MacBook Core Duo Battery Replacement
Here's the internal PRAM battery: MacBook PRAM Battery
Here's how the PRAM is replaced: MacBook Core Duo PRAM Battery Replacement
Apple Original OEM Battery. Replace your dying or broken battery with this OEM replacement. Compatible with all 13.3" MacBooks (excluding MacBook Unibody and MacBook Air). 4900-5200 mAh, 55 Watt Hours (Wh). White and black options available. — 36 available at 79.95 each.
Keeps track of dates and other system information while computer is powered off — Currently out of stock.
If you are talking about the Backup Battery (CMOS battery if you are familiar with PC's), yes. On your model, it's under the logic board.
Apple part number for hte replacement is 922-7369.
If you are talking about something in the main battery, i do not recommend taking apart any Li-ION batteries as it can be dangerous.