At 6 volts, that thing is pulling almost 1 amp of current (830 mA) under -zero- load. Four AAA alkaline batteries in series will only supply (at max, with a quality battery - you don't mention your battery brand, so I can't look up the datasheet for it) around 1200 mAh:
...and if you are using really crappy ones, it could be much less (250 mAh!). Also - you don't note what -kind- of AAA batteries you are using (so I assumed alkaline); so - for instance - if you were using NiCd or NiMH, which have a different chemistry - you wouldn't get 6 volts from 4 of them in series; you would only get about 4.8 volts.
Basically, I am saying a couple of things here:
- You haven't given us enough information to know for certain what the issue is.
- That said, most likely you need to switch to a battery cell with more capacity, likely a C or D cell (or parallel smaller cell packs together).
Also - if you are using rechargeable cells, you should use 5 of them in series, not 4, to get the required 6 volts (and in that case, you could possibly stick to using AA cells instead of AAA cells; or compromise with sub-C cells).
Finally - you don't mention how long you expect the servo to run, or what kind of load it will be under (stall current is not mentioned in the specs, but I bet its more than an amp or so); knowing this is important to size the battery properly...