The acquisition expands the Cincinnati, Ohio-based firm's geographic footprint and capabilities.
Convergys now has 125,000 employees working out of more than 150 locations in 31 countries, speaking 47 languages, it said in a statement.
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The company has a significant presence in India with offices across New Delhi, Bangalore, Gurgaon, Hyderabad, Mumbai, Pune and Thane.
When combined, total company revenue is expected to exceed $3 billion, creating the second largest customer management services provider in the world.
Stream's complementary client portfolio will also diversify Convergys' client base with addition of technology, entertainment and communications clients, it added.
"We believe our shared focus on exceptional client service, operational expertise and world-class talent will enable us to quickly integrate and leverage our expanded reach for the benefit of clients, employees, and shareholders," Convergys President and CEO Andrea Ayers said.
In January, Convergys said it would acquire Stream for $820 million in cash from funds managed by Ares Management and Providence Equity Partners, as well as from LiveIt, the BPO investment arm of Ayala Corp.