Couple things to check: 1. Battery polarity (already mentioned), 2. Measure the voltage under load i.e., with the unit turned on. Just measuring open-circuit voltage doesn't give a true reading of battery condition. You may have to slip a piece on aluminum foil under each of the battery terminals to take the reading, but that's the only accurate way. Assuming it takes 4 AA or AAA cells in series, place the foil so that you read the whole series string; not just one battery. 3. Check batteries again. Even "new" ones may be bad. 4. Does your power supply (wall wart) have the same output voltage rating as the battery pack? If you have 4 AA or AAAA cells in series, the voltage would be around 6 volts. If the wall wart also says " 6 volts d.c.", the problem may be in the wall wart connector going into the SPIII. The jack will usually have a switch contact that disconnects the batteries from the circuit when external power is used. This is so non-rechargeable batteries are not being recharged (which they can't) while connected to external power. Inspect the jack and find the switch contact that closes when the power plug is removed, and carefully clean the contact with some 600 grit emery paper. 5. Make sure the wires from the battery pack are properly soldered into the circuit board.
As an aside, I stopped in at Garmin in Olathe, KS the other day because my 2610 wasn't responding to the IR remote control. I went to the main receptionist (not the repair desk) and told her my story. She called an engineer who came about 5 minutes later and took the unit to be tested. They checked everything out and determined it was a bad remote control which they replaced free of charge. I even got a cup of coffee and a doughnut out of the deal!
Good luck with your troubleshooting, and let me know if you have problems with my instructions.