The deal, which was approved at a court meeting and a general meeting, values the company at over £500m, with 168 pence per share.
Passing with 99.8% of shareholder votes, the acquisition will see Bidco, a newly-formed limited partnership indirectly owned by investment funds advised by affiliates of Blackstone, assume control of the REIT.
The trust is scheduled to see its shares delisted from the London Stock Exchange by 22 June, with a delisting in Johannesburg expected on 27 June.
Industrials REIT said the offer represented a premium of 42.4% to the trust's price of 118p per share on 31 March, and 40.6% to the trust's one-month volume weighted average share price of 119.5p as of 31 March.