The announcement came at an NBA Board of Governors meeting Thursday.
Stern told owners during their two days of meetings of his plans, and the board unanimously decided Silver would be his successor.
"I decided that things are in great shape and there's an organization in place that will ultimately be led by Adam that is totally prepared to take it to the next level," Stern said.
Stern, who turned 70 last month, became commissioner on Feb. 1, 1984. He has been the NBA's longest-serving commissioner, establishing the league's brand around the world, presiding over team expansion and overseeing the establishment of the WNBA and the NBA Development League.
"You'll be remembered as the best of all-time," Silver told Stern, sitting to his left on a podium during a news conference.
Stern said he decided on his plans about six months ago, having guided the league through a lockout that ended nearly a year ago. He said the league is in great shape and he's confident in Silver, who has been the league's No. 2 since 2006.
"I don't know what else to say other than to recite what I told the owners yesterday in executive session," Stern said. "I told them that it's been a great run, it will continue for another 15 months, that the league is in, I think, terrific condition."
Stern is the one who got it there, taking over what was a second-rate league with little-to-no TV presence and making basketball one of the world's most popular sports.
He cited the success of the 1992 Dream Team, which helped spark the league's international explosion, but said everything "just keeps getting better than that."
"For the most part it's been a series of extraordinary experiences and enormous putting together of pieces of a puzzle and it goes on forever," Stern said. "And there will always be another piece of the puzzle and so the question is at what point do you decide that, let someone else do it? That's the point that I'm at now."
Stern was the league's outside counsel from 1966-78, then its general counsel before becoming executive vice president of business and legal affairs from 1980-84. He replaced Larry O'Brien to become the league's fourth commissioner.
The NBA has added seven franchises during his tenure.
Minnesota Timberwolves owner Glen Taylor, the outgoing chairman of the Board of Governors, said the average player salary had grown from $250,000 when Stern took over to $5 million.
The league has reported huge increases in ticket and merchandise sales, and TV ratings are at an all-time high.
"There are all kinds of other business metrics we could look at that would define David as one of the great business leaders of our time," Silver said.
Taylor and Spurs owner Peter Holt, who is replacing him as board chairman, said the owners will work to have a contract with Silver by April. Silver came to the NBA 20 years ago and served a variety of positions before becoming the deputy commissioner in 2006.
Stern said he wouldn't leave until he knew there was a successor ready, and he has repeatedly said Silver is ready for that role. Stern said he would always remain available to take a call and help the league.
"Life is a journey and it's been a spectacular journey," Stern said. "Each step along the way there are things that you have to do, things that you maybe wish you hadn't done. But I don't keep that list, and so I'm totally pleased and I'm particularly pleased with the transition of which we're now embarking."