My guess would be alternator. Check the connections on it but since you replaced the battery (assuming it was putting out a good 12 volts when you put it in) I have a feeling you'll need to rebuild/replace the alternator.

I'd suggest testing to see if the alternator is putting out anywhere near the right amount of juice but since the car won't stay running...

If you find out that it is in fact the alt I picked up a rebuilt Denso for $160 total online. PM if you want the location.