|It was acquired by the Kuwait Finance House from Bahrain in 2008.
Cape provides support services to the energy and mineral resources sectors.
Latest figures show Motherwell Bridge made a profit of nearly £5m in 2012 on
revenues of £34.6m.
The current chief executive, Russell Ward, will be retained.
He described the £37.8m deal as "excellent".
He added: "While the business has achieved good growth over the past few
years, Cape's significant resources and footprint will help accelerate and
realise our growth plans in the UK and overseas.
"There is no doubt that having the backing of a leading industry player in
the form of Cape will signal a major transformation of the company."