Thousands had patiently queued up since morning to enter the stadium but then Bio's supporters tried to force their way through passages earmarked for cars of invited guests, sparking a police charge and a stampede.
"I can only confirm one dead but more people are injured," a police official said, speaking at the ceremony which drew guests like former football legend and current Liberian President George Weah and the leaders of Senegal, Mali and Togo.
"The death figure may rise because we are still taking injured people from the stadium," said Red Cross chief Unisa Carew.
An AFP correspondent said there was chaos at the stadium after the stampede but the ceremony went through nonetheless.
Bio took office in early April after a tumultuous election campaign, ending a decade-long rule by the All People's Congress (APC).
A former soldier who briefly led a military junta more than two decades ago, Bio got 51.81 percent of vote in a runoff with ruling party candidate Damura Kamara who got 48. 19 percent.
Sierra Leone's economy is still recovering from war and disease and remains fragile with investors slowly returning, while corruption is widespread.
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