M.B.
Ok, this gets people all riled up. But I personally know of two people whos babies have died from them sleeping with them. I know you say you are a light sleeper, so am I, but I know there are things I could miss and would never take that chance. I am not trying to discourage your baby sleeping in your room, just not in your bed. My son is 8 months old and still sleeps in his pack n play in my room. I will confess, this is completely my fault, no excuses but he is my last baby and I am having troubles letting him go. Which he NEEDS to! SO im working on that! But it is very possible that she is cold, not just wants you to hold her. At one month old she is not capable of manipulating you yet or having seperation anxiety. If you want I think you should swaddle her up good and put her in that little co-sleeper you probably spent big bucks on. Mainly, just for her safety. I think you may be having a little last baby syndrome (I made that condition up myself) because I know I am, so Im not pointing any fingers. Good luck!
P.S. My baby boy is WAY more clingy than my daughter ever was.