If you want big law, UIUC. No question. It's been a top 25 program (save this year) for quite a while. They have great placement in Chicago because of the massive alumni network. If you are in the top 1/3rd, you'll be fighting alongside the Northwestern, Chicago, and Michigan students for the jobs at Skadden, Kirkland, Sidley, and Seyfarth. Ave Maria is a huge gamble. I'm currently working in Chicago and worked at a Chicago firm as an undergraduate for four summers. I never once came across an Ave Maria grad. I'd bet even less people know about AM in DC or NYC. Good luck.
Actually Ave Maria is known and well-respected in DC and any gov job for that matter. Ave Maria also places well in JAG and is now starting to do well in the biglaw DC, NY firms.