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 deal comes with £705m of net debt and potential damages claims related to the Clutha helicopter crash, in which Police Scotland's helicopter leased from Bond crashed into a 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."