The unmanned cargo ship was launched from Kazakhstan on Tuesday, but contact was lost with it soon afterwards.
Russia's space agency says Progress M-27M is now orbiting in an uncontrolled spin.
It cannot reach the ISS and is likely to break up on re-entering Earth's atmosphere.
The craft carried on to a higher orbit after contact was lost.
Igor Komarov, the head of Russia's space agency, Roscosmos, said: "The craft's continued flight and its docking with the ISS is not possible."
Two astronauts spending a year on the ISS, Scott Kelly and Mikhail Kornienko, told the Associated Press news agency that flight controllers had given up hope of controlling Progress again.
Progress is likely to re-enter Earth's atmosphere some time between 5 and 7 May.
Other capsules - including one Progress craft last year - have burned up and disintegrated on re-entering Earth's atmosphere without posing a threat to the planet's population.
Cathleen Lewis, a specialist in Russian space programmes at the Smithsonian Institution's National Air and Space Museum in Washington, said it was not yet known where debris will fall - but that it was highly unlikely to hit urban areas.
"I always remind people that two-thirds of Earth's surface is water," she said.
"The likelihood of it coming down and hitting someone is so remote as to be minuscule."
The problem with the loss of the Progress is that it puts further pressure on the volume of stores held on the space station.
ISS managers like to keep a good six months' supply of food, water, fuel and other resources in reserve at all times.
Currently, however, this volume is somewhat depleted - in part because of the explosion on launch last year of the American Cygnus freighter.
The responsibility for keeping the stores topped up leans heavily on Progress and the second American freighter, Dragon. The latter is due to make its next supply run in June.
As long as it keeps working well, ISS officials in Houston and Moscow will not be unduly concerned.
If, for some reason, supplies did become marginal, the space agencies would simply reduce the size of the normal six-person crew until matters could be improved.
Progress was to deliver food, water, fuel, oxygen and clothing to the crew of six people on the space station that orbits about 420km (250 miles) above Earth.
Even after Progress' loss, the astronauts have enough supplies to keep them going until the next expected delivery on 19 June.
A Roscosmos spokesman told Reuters that the loss was valued at 2.59 billion roubles ($50.7m, Â£32.9m).
A Russian spacecraft delivering supplies to the International Space Station (ISS) is out of control and will fall back to Earth, officials say.