I have a valid American visa lasting till June 2015 but my current passport expires in June 2013.
Does the American embassy transfer visas onto renewed passports or am I expected to travel with the old one?
Your embassy should be able to endorse the two passports for you. It's really easy. You just have to tell them to please endorse two passports together because you still have a valid US visa.
My mom has done that for years.
Here's what to do:
Having the visa in another passport does not affect it by any means, as long as the page with the visa is not clipped. I have my own US visa in the old passport and things are fine.