I know this is going to be a little useless, and bringing up a dead post, but it might just breath some air into buying old parts now that they're much cheaper now. (The Turion II Ultra M620 can easily be found for <$20 Canadian. but the M640 is $60, and I can't seem to find M660s)
I just happen to have the same model laptop but with an Athlon II M320.
This Socket S1 may be compatible with chipsets within its same class, the S1G3. Ideal processors within this class are the Turion II Ultra M500-M560 and M600-M660 series. Please note that you may also need to update the stock memory from PC2-5300 (DDR2 667) to PC2-6400 (DDR2 800)
Newer Socket S1, S1G4, typically support DDR3 SO-DIMM ram which is incompatible both by Chipset and CPU.
TDP for both the Turion II Ultra M600 and ZM84 are 35w, its the manufacturing die size that was listed at 45 and 65nm.
Although the Chipset listed by Gateway indicates its an 880G, it is not a true 880G. its more inline with specs of the M785(RS880M).