The Swedish quartet said the new material was an "unexpected consequence" of their recent decision to put together a "virtual reality" tour.
"We all four felt that, after some 35 years, it could be fun to join forces again and go into the studio," the band said on Instagram.
"And it was like time stood still."
No release date has been set for the new songs - but one of them, titled I Still Have Faith In You, will be performed in December on a TV special broadcast by the BBC and NBC.
Abba's spokesperson Gorel Hanser described the new songs, saying: "The sound will be familiar, but also modern."
The studio sessions were "like old times", she told Swedish newspaper Aftonbladet.
"Easy as anything. It didn't feel weird that they hadn't been in the studio together for 35 years."
But Hanser said the group would not perform live, other than as holograms in the forthcoming Abba Avatar tour.
"No, you can not expect them to join forces on stage again," she said. "They will not do that."
The band have resisted pressure to reform since they stopped recording together in 1982, despite a reported $1bn (£689m) offer for to tour in 2000.
In an interview with the BBC in 2013, Agnetha Faltskog said she preferred to leave the band in the past.
"It was such a long time ago, and we are getting older, and we have our different lives," she explained.
News of the new material comes in a bumper year for Abba fans. An immersive exhibition based on the band's career is running on London's South Bank, while Chess, the musical Bjorn Ulvaeus and Benny Andersson wrote with Sir Tim Rice, is being revived in the West End.
A sequel to the film version of Mamma Mia!, starring Amanda Seyfried, Lily James and Cher, will be released on 20 July.