If memory serves me right, they email you a copy of your chat after it is terminated.
I usually just select the text of the chat [CTRL-A, CTRL-C] and paste it into a Word document.
I've done three or four and that's never happened.
Actually I dont believe they email it to you, but you can email it to yourself! When you are in the chat window, there is a printer icon on the bottom right that you can click on to either print out the convo, or put in your email address and you will get a copy of the convo sent to your email. Hope it helps!