The Giver feels lighter after giving his memory to Jonas. He says, "'even transmitting that tiny memory to you - I think it lightened me a little,'" (p. 82). The memory of the snow is now in Jonas's possession, and the Giver no longer has that memory. As he describes it, the transmission of the memory has given him "'a little weight off this old body'" (p. 82).
This is the process of the memory transmission: The Giver will provide the memories for Jonas, and once Jonas has them he will begin to feel the weight of them while the Giver will feel lighter for the loss of them.