I called Xbox just now. They said 1.5 MbPs is just fine for Xbox live.
This does not mean you will have no lag at all though. It is all based on how the person who is hosting the game (I am talking about Black Ops) can connect to your internet. Since you only have 1.5 MbPs the likelihood of you hosting a game is very slim.
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if you, don't have anything else pulling any bandwidth and its a WIRED connection, providing you're getting the full 1mb, then no.
try to upgrade if possible if not under the previous rules it should be ok for online play
Be aware, the bandwidth speed the ISP tells you is the MAXIMUM speed you will ever get. So it will probably average around 10% slower depending on the time of day/evening with cable. Closer to 15% loss with DSL. On that, we don't have any issues on our X-Box with 1.5Mbps DSL IF no one else uses it to download or any other high bandwidth use.
Also note that wireless, even if you have the old "b" speed, is still faster then most internet speeds so wireless shouldn't be an issue. (of course you shouldn't be 100 feet away from the router and expect the same speed as being 10 feet away.)