You will have some contention during the restore or attach process to the new server (this should not impact the existing server). Without knowing whether or not you have a busy OLTP system or what the current performance issues are (you mention it's not RAM or CPU), then it's really hard to tell you much further information.

You will have a down time for the database being moved....but a 10GB database really isn't going to take long to move (unless your network is connected with 100MB pipes and not gigabit :-))