The 30-year-old Scot, Andy Murray's older brother, will take top spot in the doubles rankings on Monday, 4 April, replacing Brazil's Marcelo Melo.
Murray will be the first British world number one since Virginia Wade achieved the honour in doubles in 1973.
However, no Briton has previously topped the computer rankings introduced by the ATP and WTA during the 1970s.
Read more: How Jamie beat Andy to the top of the rankings
Murray gained his first Grand Slam men's doubles title in January, winning the Australian Open alongside Brazilian partner Bruno Soares.
Melo has relinquished his world number one spot after losing at the Miami Open on Sunday.
The Brazilian's failure to reach the quarter-finals means he will drop enough points to fall behind Murray, even though the Briton had already been eliminated from the tournament.
"Last night I went to bed wondering if that was the closest I would ever get," wrote Murray on Instagram. "Today driving in the car my phone started to go crazy. #1"
After losing his top ranking, Melo said: "He really deserves it. He made the final in Wimbledon, final US Open, won the Davis Cup, won the Australian Open, so he deserves it a lot. I'm happy for him."
Computerised rankings were introduced by the ATP for singles in 1973 and for doubles in 1976. The WTA introduced computer rankings for singles in 1975 and doubles in 1984.
Before that, world rankings were compiled by leading tennis journalists and issued annually. Fred Perry topped the men's world rankings in the 1930s under this system, while fellow Briton Angela Mortimer was the women's singles number one in 1961.
Jamie Murray will make history as the first Briton under the modern ranking system to become a world number one.