The problem with a $1 buy it now listing is that a third party can inadvertently hit the buy it now and end your listing. Chance is small, but you'll be in a mess if that ever happens.

However, you can create a regular 7-day auction listing starting at $0.01 with free shipping, have your buyer bid on it, and then you can end the listing by selling to the highest bidder. If a third party happens to bid on it, you can always cancel their bids, leaving only your intended buyer's bid of $0.01. And then you can end your listing by selling to the highest bidder.