Avincis, which operates as Bond Aviation in the UK, provides helicopter and fixed-wing services for medical, search and rescue, firefighting and civil protection in Europe and offshore crew-change services to the oil and gas industry in the North Sea.
The company is seen as a "strategic and cultural fit" for Babcock, which supplies engineering services in the defence, energy, telecommunications, transport and education sectors.
The £920 million acquisition comes with £705 million of net debt and potential damages claims related to the Glasgow helicopter crash in November, in which Police Scotland's helicopter leased from Bond crashed into the Clutha pub, killing 10 people.
Babcock, which runs Rosyth dockyard, has also advised that the prospect of Scottish independence could have "adverse" impacts for the newly enlarged firm.
Chief executive Peter Rogers said: "Avincis already has a strong growth platform and its combination with Babcock will generate even greater expansion opportunities and value creation for Babcock's shareholders."
James Drummond, chief executive of Avincis, said: "Babcock is a great strategic and cultural fit for Avincis.
"We provide vital, complex and life-saving services for our customers, and both share a commitment to operational excellence and the highest standards of safety. Avincis's global business has grown significantly over the past few years, and joining forces with Babcock will allow the strengthened group to access new markets and customers, and build on an already impressive order intake."
Babcock's acquisition announcement stated: "Bond has received several intimations of civil claims for compensation for personal injury and property damage from solicitors acting for people impacted by the (Glasgow helicopter) accident, including, without limitation, a potential employers' liability claim in respect of the Bond Air Services pilot who died in the accident.
"While waiting to know the final outcome of these investigations, Bond's insurance company has informed it that, without any acknowledgement of liabilities, the insurance company has started to make certain interim payments to give support to some of the people impacted by the accident that had intimated a civil claim."
It added: "If Scotland becomes independent, there is likely to be a lengthy period of uncertainty in respect of the new Scottish government's policies and their impact, some of which can be adverse, on the Babcock Group's and, following the acquisition, the enlarged group's Scottish businesses.
"There may also be a medium term knock-on effect on the nature, timing and scope of the policies and procurement plans of the successor British state, especially in defence terms."