The deal means that Gresham will buy the business from current owner, equity firm Permira, for a sum in excess of £110m, including £35m in equity that is being funded by the Royal Bank of Scotland and The Bank of Scotland.
Betts is one of the leading global players in the toothpaste tube packaging markets and supplies manufacturers such as Colgate Palmolive, GlaxoSmithKline, Procter & Gamble and Unilever.
Permira acquired Betts nine years ago for £82m through a management buyout.
At the time the company was trading under the name Courtaulds Packaging.
The deal aims to boost the company's capabilities in the fast-growing market for laminate tubes, where growth is particularly strong in the emerging markets.
The company has 1,300 employees worldwide, with 14 manufacturing sites in eight countries, including the UK and the USA.
However, the company has expanded significantly in key emerging markets, with other sites now located in Mexico, China, Brazil, Indonesia, Poland and India, and it is in these markets that the company is expected to focus its new investment efforts.
Paul Batemen has been appointed chairman of Betts.
He was previously operations director for Boots and brings his experience from the manufacturing, HR, supply chain, HR, logistics and IT systems.
"Our business is well poised for further growth in our market," said Bateman.
"The support and financial firepower of Gresham will enable us to invest in our manufacturing facilities.
This will provide the resource and capability to support our customers' needs and undertake further expansion in the future."