Mr Desmond had been One51’s largest shareholder, with his investment represented by his International Investment and Underwriting business and certain other related parties.
As a result of the move, IIU’s non-executive representative on the One51 board, Pat Gilroy, has resigned.
One51 has previously worked with CDPQ, having partnered on the Dublin company’s purchase in 2015 of North American plastic products manufacturer IPL.
One51 — whose other notable shareholders include beef baron Larry Goodman and other various co-ops — said earlier this year that it was eyeing further acquisitions in North America and mainland Europe as it continued to sharpen its focus on its core plastics operations.
IPL, which has six manufacturing plants in Canada and the US, makes products for the food packaging and logistics industries.
One51 owns a 67% stake in IPL and CDPQ holds a further 22%.
One51 recently sold the Irish and UK arms of its ClearCircle environmental services business in separate transactions.
It had long been thought that the purchase of Mr Desmond’s stake could pave the way in time for an initial public offering in One51 shares.
Mr Desmond increased his stake in One51 in August 2015 — by buying out fellow shareholder Pageant Holdings.
Private equity group CapVest had been linked with a €282m bid for One51 at the time.
Shares in One51, which trade privately, were worth €1.80 each. They are up from €1.35 at the start of the year, according to the company’s website.