Answer:
The author doesn't tell us what the real ending of the story is. It is possible that when Lowry wrote The Giver, she had a sequel in mind. Leaving the ending of this book ambiguous would encourage readers to purchase the sequel. You can read Gathering Blue and Messenger, which were written as companion books to The Giver and will give you more information.
Going on just what the book says, there are two conclusions you could reach.
First, Jonas and Gabriel die. The book makes it clear that they are slowly freezing to death. They are weak, hungry and tired. The book also says that Jonas uses his last little bit of strength to find the sled waiting for him at the top of the hill. They sled down the hill to "Elsewhere", perhaps an afterlife of some kind that follows death. This could be why Jonas heard music as he slowly slipped down the hill. In addition, the Giver was transmitting memories to Jonas before he died, giving the idea that if the Giver died before they were transmitted, they would be lost forever or else freed and allowed to enter the minds of the people. Jonas, now, being the new keeper of the memories, would release those memories when he died. This could be why he also heard singing behind him - as he slipped away, his friends and family received the memories he had carried and were freed from the austere existence they had experienced.
Second, Jonas and Gabriel find "Elsewhere", an unexplained real and literal place. This is supported by the fact that there is a sled waiting for them, apparently placed there by people who are hoping he will find it and use it. It would only stand to reason that if they placed it there for Jonas, they would be waiting for him at the bottom of the hill. This is confirmed near the end of the chapter when it says that he knew they were waiting for him and the baby. The book also supports this idea because he heard music and saw lights and warmth coming from Elsewhere, indicating that there is life and emotion there. There is also the possibility that his leaving freed the residents of his home town and enabled them to have memory, which is why he heard them singing behind him.
I would suggest that the author wanted you to come to your own conclusions, so she intentionally left it very ambiguous. Reading the two companion books mentioned above would help if you simply can't stand ambiguous endings (like me!).
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