I'm not quite sure why you would want to snap and entier poly to a curve but there could be some use i'm sure, though the usualy modeling method is to snap the Verts to the curve. to do that you need to hold the "C" key. then cut the edges into the mesh once its finished.

I find its a little harder to snap the points to the curve after the shape has been completed, it might be an error or something on my end but I usualy try to make the shape rite the first time rather than forcing the edges to cooperate after the fact.

Make sure that you set up the object so its modeling with Quads. you can always reduce later.