i'd say it depends on a few things. you don't want to leave your current employer in a bind, is there others who can pick up where you left off?
i guess what i'm saying is to avoid burning a bridge if possible but always look out for #1 first. try to appease both sides if possible, if not, then make the new employer happy while trying to not make current employer too mad.
2 weeks is standard, a lot of employers accept 2 weeks and walk you out after 1 day. if this was the screening interview or 1st interview, you've got a long way to go. also tell them you can get moved quicker if they help financially, temp housing/points on house/moving expenses, etc.