The company announced the deal in March in a bid to strengthen its position in India’s competitive mobile market.
The combined business will be renamed Vodafone Idea Ltd and will remain listed in India.
Aditya Birla Group, the majority shareholder of Idea Cellular, is to buy a 4.8% stake in Vodafone Idea from Vodafone for INR26bn (£281mln). Following completion of the deal, Vodafone will own a 45.2% stake in Vodafone Idea and Aditya Birla will own a 26.0% stake.
Former Vodafone India chief operation officer Balesh Sharma has been appointed the chief executive of the merged business.
In the year to June 30, Vodafone India and Idea generated revenue of INR585bn (£6.3bn) and earnings (EBITDA) of INR107bn (£1.2bn).
Vodafone said the merged company is expected to deliver INR140bn (£1.5bn) in run-rate cost and capex synergies.
The news comes a day after Vodafone said its Australian business and local rival TPGTelecom Limited have agreed to merge into a single A$15bn telecommunications company that will be able to better compete with Telstra and Optus.
Fierce competition in some of its markets, including a price war in India, led to a slowdown in first-quarter organic service revenue.