Ford Escort won't start in the cold

So I'm having a problem with my 2000 Ford Escort. I think it may be the cellenoids in the starter, but I'm not sure. It won't start when its cold outside. It starts relatively fine when its warm or when its been in the garage. When its been outside for a short time, I'll turn the key and after about 3 seconds the engine will kick and start easily. When its been in the cold for more than about 12 hours though, it won't start at all. Any ideas? Also, any hints on what I could do to not have to replace the starter? Would a blanket over the battery when its not in use help at all?

Thanks