Alkass Digital
The NBA draft is made up of two rounds of 30 picks, with 30 teams choosing players based on where they finished last season.
The worst performing teams pick at the top of the draft, where they have access to the best incoming talent.
The 2023 draft was held in Brooklyn with NBA commissioner Adam Silver announcing the picks.
"This is accomplishing something that I have been dreaming of my whole life," Wembanyama told ESPN.
"Hearing that sentence from Adam Silver, I've dreamed of it so much that I have to cry, man."
French president Emmanuel Macron congratulated Wembanyama on becoming "the first Frenchman to be the first pick in the NBA draft".
In a post on social media, he added: "You're already making us dream, Victor Wembanyama. No doubt about it: you'll change the game!"
Wembanyama played for Boulogne-Levallois Metropolitans 92 in France during the past season and averaged 20.6 points, 10.1 rebounds, 3.0 blocked shots and 2.5 assists in 44 games.
"We're thrilled that we were able to bring Victor on board," said Spurs coach Gregg Popovich. "He's obviously a heck of a talent, a very mature young man.
"But just like with every draft pick, whether it's the first pick or the 27th pick, or the 38th pick, we have a responsibility to each and every one to try to create an environment where they can reach the best success possible for them.
"We're going to get him on the court and we're going to see him play, and we'll go from there."
How good is 'alien' Wembanyama?
LA Lakers star LeBron James watched when Wembanyama starred in an exhibition game in Las Vegas last October.
"He's more like an alien, he's for sure a generational talent," said James.
In basketball circles the teenager is what is known as a 'unicorn' - a player who can do it all - but Wembanyama recently said he preferred the description from James.
"First of all, I'm really glad he said that because I didn't like to be called a unicorn," Wembanyama told Sports Illustrated. "I like it because it's just something not from this world.
"I like being called an alien, yeah. It's really what I'm working to be - something unique and original."
Dr Lindsay Sarah Krasnoff, author of the forthcoming book Basketball Empire: France and the Making of a Global NBA and WNBA, spoke to BBC World Service Sport programme Sportshour about Wembanyama.
"He's got the speed for the game, he's got the technical finesse of the game as much as any 19-year-old could be and then some," she said.
"He's got the athleticism but also, really importantly, he's got the basketball IQ. He can read the court pretty well.
"He's got a good sense, not just of the court, but where his team-mates are and just the way that he's able to move that tall, lanky body in ways that you would not typically associate typical basketball big centres doing, it's just amazing."
Brandon Miller, a 6ft 9in 20-year-old who starred at the University of Alabama, was the number two pick in the draft and went to the Charlotte Hornets.
Scoot Henderson, who has played the past two seasons in the NBA's developmental 'G-league', was chosen by the Portland Trail Blazers as the third pick.
The 20-year-old identical twins Amen and Ausar Thompson made up the top five selections.
Amen Thompson went to the Houston Rockets as fourth pick in the draft, while his brother Ausar was picked fifth by the Detroit Pistons.