More than 50 fire fighters battled the blaze at the bar in the city centre, said French Interior Minister Bernard Cazeneuve.
He gave an initial count of 13 dead and six injured in the fire, which started late Friday night or shortly after midnight.
A judicial inquiry has been opened to determine the cause of the blaze, Cazeneuve said in a statement.
The local Paris-Normandy newspaper said the fire ignited in the basement of the Cuba Libre bar, which was rented for the evening for the birthday celebration, AP reports.
The paper, without citing sources, said the ceiling quickly ignited when candles were lit.
A tense France is on maximum terror alert after two attacks last month, the latest the killing of a priest in his church outside Rouen.
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