(Possible spoilers for mission 1)

My game runs 100% fine, right up until I am about to finish the first mission. Then once the final borg sphere is destroyed, it seems to run dialog from Captain Picard as well as launch the cut scene all at once. Then it all cuts out, and either dumps me to desktop or launches into mission 2.

I've had this issue on a stock install, as well as with unofficial patch 1.3, it doesn't make any difference.

Has anyone encountered an issue like this before?

Open the dxconfig Tool inside the game directory and set your Resolution, you need to set your Refresh Rate to 60 Hz above may cause issues such as the double audio.