Hartley joined Toro Rosso in late 2017 as Carlos Sainz moved to Renault but struggled to match team-mate Pierre Gasly.
The New Zealander spent much of 2018 defending himself against speculation he was on the verge of losing his drive, with Lando Norris touted as his replacement before McLaren blocked the move.
Albon has long been linked with the seat but a contract with Nissan to race in Formula E had initially blocked the move.
Nissan and the e.dams team confirmed they had split with Albon, opening the path for Toro Rosso to confirm the British/Thai driver.
“Nissan e.dams and Albon have agreed to part ways with Albon exiting his contract to take up a different opportunity in motorsport,” a statement from the Japanese manufacturer confirmed.
It was swiftly followed by one from Toro Rosso, confirming what is in effect the final available seat on the 2019 grid.
“Alexander had an impressive Formula 2 season in 2018. He won four races and finished the championship third,” said Toro Rosso team boss Franz Tost.
“The way he is able to overtake many of his rivals in the races shows that he is ready and matured to race in Formula 1.
“Scuderia Toro Rosso is very much looking forward to 2019, as with Daniil and Alex we have two young, very strong and competitive drivers.”
Albon added: “It's such an amazing feeling to know that I'm in Formula 1 next year.
“Throughout my single seater career, I went through a few ups and downs. I was dropped by Red Bull in 2012, so from then I knew my road to Formula 1 was going to be a lot harder.
“I worked really hard and tried to impress every time I got in the car, and I have to say a big thank you to Red Bull and Dr Marko for believing in me and giving me a second chance.
“I've always been motorsport mad and since I first got in a car it's been my dream to be in Formula 1. To be given this opportunity is just incredible.”
Albon will join Daniil Kvyat at the team next year in what will be the Russian's third stint with the team.
With Lance Stroll all but confirmed at Force India there are now no seats to be accounted for in 2019, making it highly unlikely that Hartley will appear on the F1 grid next year.
In 28 grands prix, the 29-year-old amassed four championship points with a best finish of ninth at the United States Grand Prix.