The discovery came as the Northern Fleet's chief-of-staff Mikhail Motsak said the navy's "worst fears" had been confirmed.
The Kursk went down in the Barents Sea after suffering a catastrophic explosion during naval exercises on 12 August.
In the Russian press, President Vladimir Putin has come under increasingly fierce criticism, and an opposition party has called for a full parliamentary inquiry.
First body found
The first corpse of a sailor from the Kursk was found on Monday close to an escape hatch, Russian television reported.
The discovery came as the Northern Fleet's chief-of-staff Mikhail Motsak said the navy's "worst fears" had been confirmed.
"All the compartments of the submarine are flooded with water. None of the crew are still alive," he said.
Norwegian officials said that Russia had asked Oslo for help in recovering the bodies of the crew.
Earlier on Monday the Russians said a British rescue craft at the scene would not be used in the operation as the escape hatch was too damaged to provide an airtight connection.
Russian television has reported that divers have detected no evidence of any radioactive leaks.
The divers have been working from a diving bell 108 metres below the surface, wearing special suits to protect them against the extreme depth and cold.
Raising the Kursk
Russia is planning an international effort to lift the Kursk from the seabed, Russian deputy prime minister Ilya Klebanov said.
"Not a single country on its own can handle such an operation," he told Russian television.
He said initial plans to lift the submarine with pontoons would be ready in three weeks.
The BBC's defence correspondent Jonathan Marcus says the relative speed with which the Norwegian diving team went into action and managed to open the submarine hatch only adds to questions surrounding the Russia's own rescue efforts.
Call for answers
Amid growing anger in Russia over the failed rescue operation, there has been a call for a full parliamentary inquiry into the disaster.
A leading opposition party, Yabloko, said the delays might have cost lives.
"The fog that was built up around this tragedy must be dissolved," said MP Sergei Ivanenko.
"Society should know the answer to the one question that still stands before people: Was everything done to save people?"
The speaker of parliament, Gennady Seleznyov, said he believed that the military had not been quick enough in letting President Putin know the gravity of the disaster.
Correspondents say that the president must now decide whether to sacrifice leading members of the military - a move which would alienate what has been until now one of his most important sources of support.
Press fury
President Putin on Monday tripled funds for the relatives of the submarine crew to an amount equal to $54,000.
He said they had to be helped with money for accommodation, transport and telephone calls.
Extra psychologists are also being sent to the submarine's home base at Severomorsk to help counsel the families.
But one newspaper in Moscow has published photos of President Putin in shirtsleeves on holiday last week, plus the defence minister and the head of the navy.
The headline above them was: "They don't sink".
Russia has said the Kursk may have hit a World War II mine or collided with a foreign submarine.
Experts believe a torpedo may have exploded in the front section of the Kursk, triggering a much larger blast.
US and Norwegian authorities detected two explosions in the area at the time the Kursk went down.