Hi Dan,
There are a number of approaches to take to break the awkward silence. Trying using the power of being obvious....that's right, it may seem counterproductive, but it works!
When you run out of things to say during a conversation with a woman and awkward silence ensues, simply admit that you both of you are NOT great conversationalists. This way you will never be at a lack of words to say in conversation.
Trust me, for some odd reason, this actually works. This technique dissolves the tension and can even cause her to laugh and you guys can both continue the conversation naturally from there!
Good luck!
-Scott