Manuel Valls told French radio there would be "no discussion, no negotiation and we will never give in to blackmail. Even if we are of course very worried after the authentication of this video."
The group, Jund al-Khilifa (Soldiers of the Caliphate), promised in a video message to kill the hostage, Herve Pierre Gourdel, within 24 hours unless France stopped its air strikes against the IS in Iraq.
The video footage was confirmed as authentic by the French government, and came after IS issued a statement urging Muslims to kill Westerners whose nations have joined a campaign to battle the jihadist group.
On Friday, France conducted its first air strikes in Iraq but has said it will not carry out such operations in Syria.
"These strikes, this engagement will of course continue," promised Valls.
"France is a great nation which completely lives up to its responsibilities," added the prime minister.
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