"Given the number of ballots that remain to be counted, it is apparent the election has been decided and Mayor Begich has been elected," Stevens, 85, the longest-serving Republican in Senate history, said in a statement.
"My family and I wish to thank the thousands of Alaskans who stood by us and who supported my re-election. It was a tough fight that would not have been possible without the help of so many Alaskans -- people who I am honored to call my friends.
"I am deeply grateful to Alaskans for allowing me to serve them for 40 years in the U.S. Senate."
In forgoing a recount, Stevens went on to say he wishes Begich and his family well, and that he and his staff are ready to help him in the transition.
Begich had claimed victory Tuesday after the tally of absentee and questioned ballots widened his lead to 3,724 votes with only about 2,500 ballots from overseas outstanding.
Begich, 45, said Wednesday he intends to talk with Stevens, Republican Gov. Sarah Palin and others about "how we work as a team."
About a week before the Nov. 4 election, Stevens was convicted of seven felony charges that he failed to disclose more than $250,000 in gifts on Senate financial disclosure forms.