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You want to record all 1000 messages into one audio file - right? Yes, you can do it using the recording app already on your Mac, eg. Garageband. I assume though that you can simply press the Play feature in UVR, and the messages will stream out one after another.
The best recording app is WireTap (Studio), but it's not cheap. Amadeus Pro too is very good. Both will record any audio going through your Mac. The gaps between messages will also be recorded, but you can edit them out with Amadeus (or QTime Pro) later. Be sure to record in .mp3 format or the ensuing file will be very large.