Grey beat Leo Burnett and McCann-Erickson to the business, which was previously held by FCB. MediaCom will handle the global media account.
Richard Badler, Unisys vice-president of corporate communications, said: 'We believe we laid the groundwork for a strong relationship through the review process, and we will hit the ground running. We expect to launch a new creative campaign in the first quarter of 2001.'
Unisys has also appointed Leigh Alexander as chief marketing officer, based in the US. She will responsible for strategic global marketing.
Previously she was president of Comdial Enterprise Solutions, and has also held senior positions at Philips Electronics and General Foods.
In July, Unisys saw a third of its share value wiped out after issuing its third profits warning in nine months.
The company also owns a quarter of UK telecoms company Quip, which launched in February. Bank of Scotland also owns a quarter of the company, with BT taking the majority share.