Jay-Z (centre) pictured at New York Fashion week in February 2015. Photo: TT
The firm's current owners, who have 90 percent of shares in Aspiro, have accepted an offer worth $54 million (approximately 466 million kronor).
According to a statement from Jay-Z's company Project Panther Bidco all the conditions of the purchase have been fulfilled.
Aspiro has announced that the Nordic business will be delisted from the Stockholm Stock Exchange.
News of the deal came just over a week after it emerged that ten percent of Aspiro's shareholders had rejected the rapper's original bid, reported to have been $56 million.
They argued Jay Z's offer undervalued the company's future potential. It is currently the largest rival to the Swedish-owned music streaming service Spotify.
The largest single shareholder in the firm is the Oslo-based media group Schibsted, which together with Norwegian record company Platekompaniet AS and Norwegian Private Equity company Verdane Capital, owns a 76 percent stake.
At the end of the third quarter of 2014, Wimp said it had 512,000 paying users in Scandinavia, Germany and Poland.
Spotify, a pioneer in the streaming music business currently has over 15 million paying subscribers.
Jay Z's expected takeover of Wimp will see him vie for a slice of the music streaming market pie alongside other US music stars. Apple acquired the site Beats Music from another rapper Dr Dre, in a much publicized $3.2 billion deal (SEK 26.6bn) last year.
In addition to several music-related ventures, Jay Z has branched out into fashion and last year bought the champagne brand Armand de Brignac.
Married to singer Beyonce, he was listed as the third richest hip hop artist on the planet last year, by Forbes Magazine.