Why go someplace where this would be a concern. It's a first date. You're not in a relationship with this woman, hell you don't even know if there's going to be a second date.
If you feel you MUST pay, just go someplace to have coffee. Before anybody starts flaming, consider this; why go someplace elaborate? She is supposed to be impressed with YOU, not where you go or what you buy.
Unless you have low interpersonal skills or don't mind hooking up with a woman who's more into where you take her, don't try to impress her with just the venue. Yes, ambiance does make a difference, but you don't have to break the bank for it, especially for someone you just met.