|Audrey Sauret (France) had 14 points|
Miodrag Veskovic, whose side could ill afford to lose for the second straight day, beat the defending champions 70-68 as Les Bleues missed a potential game-winning shot with just two seconds remaining.
Alain Jardel's side gained possession with 24 seconds remaining and trailing 69-68.
They ran the clock down to just two seconds and Nicole Antibes, the 27-year-old centre who scored 12 points in their win over Israel on Friday, missed an open baseline jumper.
Serbia & Montenegro's Slobodanka Tuvic rebounded and was fouled. She made one of two free throws at the other end and France had no time to get off a game-tying shot.
Veskovic had been talking up his side's chances of the title before the tournament so the setback against Poland has provided an early gut check.
His team seemed to be in control at half-time after storming out to a 25-10 first quarter lead and then holding a 40-27 advantage at the break.
But France clawed their way back in the third and trailed by just four points going into the final quarter.
Nicole Lesdema's three-point play gave them their first lead with around five minutes to play.
Neither side could pull away, though France did have an excellent chance to lead by three. They forced a steal and appeared to be going in for a layup, but play had to be stopped because the shotclock had not been re-set.
Serbia & Montenegro took advantage of the lucky break by rebounding a miss, going to the other end and taking a 69-68 lead on a basket by Gordana Kovacevic.
France then failed to convert their last chance, leaving both sides with a single triumph and one defeat from their first two games.
Gordana Grubin's 17 points paced the Serbo-Montenegrins while Tuvic worked hard under the basket and grabbed 12 rebounds.
Lesdema had a game-high 18 for France, but the misfiring from Cathy Melain proved costly.
The co-captain made just two of 12 shots from the field - including one of six from three-point land - but did hit a crucial three late in the game when it was in the balance.