Here's how:

Before you close the chat tab, click the envelope icon in the upper-right corner. It’s one of three buttons above the chat window, to the right. Click it to convert your chat to an email that's addressed to the contact you were chatting with. If that person's email address is not in your Contacts, just fill out the “To:” field manually.

If or when the other person signs out during a chat, you can click send (the person's name) an email, which is another way to convert a chat to an email. That way, you won't lose any last comments.

Finally, converting a chat to an email is currently the only way to save chat text. If you want to save a chat, convert it to an email and send it to yourself!