LGE, based at the Heriot-Watt University Research Park in Edinburgh, designs and builds plants for processing and storing liquid gas.
It is a major supplier to the marine and onshore energy sectors and employs around 50 people.
For its 2011 financial year LGE had gross revenue of £26.3m and operating profit of £3m.
Babcock plans to combine the company with its design services business unit, which is part of the wider marine and technology division.
The enlarged business will offer mechanical, electrical and chemical processing equipment for ships, oil and gas platforms and onshore support facilities.
Peter Rogers, chief executive of Babcock, said: "The acquisition of LGE and its combination with our existing expertise in Design Services will create a strong business, well positioned to build on its current positions and develop into new commercial marine markets, particularly within the oil and gas market."
Weir also confirmed it has completed the £237m takeover of United States-based fracking equipment specialist Mathena, which was announced last month.