The rig market deal is Borr's biggest since it was set up last year by Tor Olav Troeim and other executives who had left Seadrill, once the jewel in the crown of Norwegian-born shipping tycoon John Fredriksen but now battling with $14 billion in debt and liabilities.
After years at Fredriksen's side, Troeim split with him in 2014.
Since then Troeim has re-established himself as an independent player in the global shipping market with a high profile and a reputation for successful capital raising.
Transocean, executives at Borr Drilling and Troeim were not immediately available for comment.
Fredriksen's private investment vehicle Seatankers bought the new West Mira rig from the Hyundai Samho Heavy Industries shipyard for an undisclosed sum just a week ago.
As the price of crude has fallen by more than 50 percent since 2014, oil firms have cut back on rig hires, leaving many vessels idle and prompting owners to restructure operations to preserve cash.
Transocean shares were down 4.8 percent at 1403 GMT, lagging a European oil and gas index down 1 percent.
Borr, founded with the aim of picking up cheap assets as rig firms sell during the industry downturn, said it would issue $800 million in new shares to help finance the purchase of the 15 rigs.
The deal comprises 10 high-specification jack-up rigs and five more that are under construction, Borr said in a statement. Swiss-based Transocean will retain a fleet of about 50 larger rigs used for exploration in deeper waters.
Borr, which currently owns just two rigs, will pay $90 million on average for each of the 15 Transocean rigs.
"That's half the construction cost, so it's a pretty attractive deal," said Carnegie brokerage analyst Frederik Lunde.
Borr, which is listed on Oslo's over-the-counter board, said a group of investors had agreed to the $800 million share issue to help fund the deal. The new shares will be sold at $3.5 each, a discount to Borr's current share price of 35 Norwegian crowns ($4).
Boor says it expects the acquisition to be completed by the end of May.
In addition to the 15 shallow-water rigs, Transocean owns 30 ultra-deepwater units and has four more under construction, as well as seven harsh-environment rigs, three deepwater rigs and six mid-water units, according to its website. Keppel Corporation, which is building the five rigs that are under construction, said in a separate statement it had agreed to transfer the ownership to Borr from Transocean.
Transocean had originally agreed to pay $219 million for each of the five rigs, Keppel said, adding that this would be only slightly reduced, to $216 million per unit under the amended deal.
($1 = 8.4678 Norwegian crowns)
(By Terje Solsvik and Jonathan Saul; Additional reporting by Nerijus Adomaitis and Ole Petter Skonnord; Writing by Gwladys Fouche; Editing by Susan Fenton/Ruth Pitchford)