Moving old posts to new blog

I followed these instructions:

In the WordPress dashboard you can go to Tools > Export > Download Export File. Then, in your new installation, go to Tools > Import > WordPress, then browse and upload the XML file which your previous export downloaded for you.

Which worked fine, except that all the images point to the old site instead of transferring over to the new site, so when I take the old site offline all the images will be broken.

Is there a way to transfer all the images? Thanks so much! I am truly stumped!