Right now the least expensive carrier in either direction is through the Postal Service.

$80-125 for frame / $175-225 for a complete bike with some tracking/signature required service.

Make sure to verify what the total dimension and dimensional weight is without incurring any additional handling fees. I would recommend using a frame box and a wheel box for shipping a complete bike. You can spread out the components between the two boxes to keep the weight down on each.

You can have the seller declare it as a "Gift" along with an HONEST declared value and you shouldn't incur any additional fees or delays with customs. Make sure both Ship From and Ship To address' are residential and not from/to a business.

A good packing job goes without saying.