The US-based owner of Carling, Caffreys and Staropramen announced the deal on Monday for an undisclosed sum.
Sources have told Sky News that Aspall, founded in Suffolk in 1728 by Clement Chevalier, is likely to have an enterprise value in excess of £40m.
Molson Coors said the acquisition "brings together two companies with a passion for building extraordinary brands and similar family-orientated ownership structures, with close to 650 years of combined experience in operating excellence".
It said it planned to grow the business in the UK by investing in the firm's Suffolk operation, as well as "establish a leading presence for the Aspall brand in key markets around the world".
It added members of the Chevalier family would remain part of the business and play a key role in "shaping the strategic direction of Aspall", which has a 1.6% share of the UK cider market.
Phil Whitehead, managing director of Molson Coors UK & Ireland, said: "We're delighted to add Aspall to the Molson Coors portfolio. Both companies share a similar history that is deeply rooted in family, dedication to customers and a commitment to excellence.
"We're now looking forward to helping Aspall become the number one premium cider in the UK and building on the huge potential of the Aspall vinegars, as part of an ongoing strategy to premiumise our portfolio."
Barry Chevallier Guild, Aspall chairman, described the deal as "an important milestone in Aspall's long history and a proud day for everyone involved with the company".
"Having been in close discussions with Molson Coors for over a year, we were delighted to find that they share our rich heritage, passion for making quality cider and vision for the future development of Aspall and its people," he said.
"Molson Coors is known for respecting the provenance of local brands it has acquired in the past, and has the scale and expertise to accelerate our growth in the premium cider category in the UK and beyond."
Henry Chevallier Guild, one of the owner-directors of Aspall, told Sky News last year that the company would only strike a deal with a partner "who wants to preserve the values and heritage of the brand".
He said of the takeover: "We believe that Molson Coors investment will provide the catalyst to grow Aspall and build the recognition for quality cider worldwide."