Cats and baby?

Anyone have a very affectionate cat and nervous about them with baby? I have a cat, a fat cuddly cat, that jumps in and on everything. All night he goes from room to Room and bed to bed sleeping with everyone. Ive woken up short of breath only to realize its because my cat is sleeping on my chest. He does the same with my husband and older son but never my five year old. Im very nervous about him jumping in the bassinet with baby and accidentally harming her. His weight on the chest of a new born could be dire. Does anyone else have this concern or ideas to keep kitty out.. I've been putting stuff in the bassinet for now too keep him out.