Yes but isn't the voltage slowly falling over that 1 hour period (in the case of a 1000 mAH battery powering a 1 amp load)? Wouldn't the voltage fall to a level that is un-useable before the hour is up? Or does the battery capacity suggest an amount of power that can be delivered before the voltage starts to sag?
Depends on the specific battery chemistry one is talking about. Nicad and nimh have pretty flat cell discharge voltage over most of the charge 1.2 volts per cell, so their terminal voltage is pretty constant till the end and they then fall off the cliff so to speak. Lead acid and lipo have a pretty linear voltage ramp down from full charged to full discharged, so while a lipo cell in nominally called a 3.7vdc cell it starts out fully charged at 4.2 and is consider toast at 3.0vdc.
mAH rating is just a crude estimate to use in estimating battery duration. Most battery manufacures let their marketing department state the mAH rating of a given battery the sell, that is why lots of people say to take their ratings with a grain of salt. A rating that would consider both current and voltage during discharge would to it's watt-hour rating as watts is a product of both volts X current.
The best acid test of how well a specific battery will power a specific project and for how long is to bench test it's performance during prototyping phase and adjust according to timed results seen.