C&WW needed 75% of votes to agree to the takeover at a shareholder meeting on Monday, after receiving some 58% of proxy votes supporting the move.
The decision comes two months after a takeover deadline for C&WW was extended, after India's Tata Communications pulled out of negotiations.
The offer valued the entire issued ordinary share capital of C&WW at over £1bn.
At the time Vodafone said the offer for C&WW gave a 92% premium to the 19.8p closing price on February 10, which was the last business day before the offer closed.
It is believed the acquisition of the C&WW fibre system would be used by Vodafone to expand the corporate client base as Britain's consumer mobile network reaches handset saturation.
The company has its roots in the 1850s after Sir John Pender joined the board of the English and Irish Magnetic Telegraph Company.
It was involved from the start in the laying of underwater submarine cables around the world, which linked the outposts of empire and provided revolutionary speed of communications.
It now owns more than 250,000 miles of submarine cables linking 150 countries but Vodafone is expected to sell off the undersea cables business.