1000w amp is a really big amp! It sounds like a voltage issue. Turning the amp up is drawing so much power that the voltage getting to the CD player falls below where it is able to function. There could be two common causes for this: 1) The power supply wires to your Amp are too small, or 2) Your battery is too small and when asked to supply the large current draw for the amp at high volume is unable to keep the voltage up. If you can, put a VOM meter on the batter (right at the battery) and see what the battery voltage drops to when you crank the volume. If it drops too much below 10.5v, then that is a problem. If it stays high, try next at the amp. If the voltage is much lower at the amp then the battery then your wires are undersized. Finally, check at the CD player its self. If you are trying to share power wires from the battery to the CD player AND the amp, that is not a good way. The Amp should have its own set of wires all the way to the battery and do not tap off anything else from these. (and btw: Make sure to run a black neg wire all the way back to the battery, don't use the cars frame to carry the neg back) If none of the above, a solution might be to give the amp its own dedicated battery, mounted say right next to the amp. (There are battery splitters that can be used to recharge this. Look at RV stores) Or, your current battery might be getting old and needing replacement, or you might be able to 'upgrade' the battery to a different type e.g. the spiral wound ones) which can supply more current while keeping the voltage up. Good luck!

If the vehicle is running, I'm thinking it's more likely the alternator that's not keeping up, not the battery. It's my (carstupid) understanding that the battery is just there to get the vehicle started and keep things running while the engine is off, but once things are started everything is powered by the alternator.

well your understanding is wrong. yes the battery is there mainly to get the car started then it doesnt do anything really. but when you hook something additional up to it its all in how you wire it up. go to you local car stereo store and buy an amp wiring kit for the watt range your looking for and follow the instructions. put a new battery in your car and make sure that if your using it for subs that the wattage of your subs is not less than your amps. and that will definitely solve your problem.