The £450m price values Channel 5 at around nine times last year’s reported earnings before tax, depreciation and amortisation.
Viacom will be the first US firm to control a British public service broadcaster.
Viacom’s chief executive Philippe Dauman said of the deal:
The acquisition of Channel 5 accelerates Viacom's strategy in the UK, one of the world's most important and valuable media markets.
Channel 5 is known for broadcasting shows such as CSI: Crime Scene Investigation, Men In Black, Big Brother and children’s programmes like Peppa Pig.