|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.
Miodrag Veskovic, Head Coach, Serbia and Montenegro
"We are very happy to have beaten the European Champions, and this was a special victory. France is a good team and you can learn a lot from the way they play basketball."
"The only way for us to win was to play good defense and rebound, and that is exactly what we did. We also had some luck in the final moments of the game."
"This win will give us strength for the rest of the competition."
Slobodanka Tuvic, Serbia and Montenegro
"We had a tough game yesterday against Poland and we are happy to win. We also have another hard game tomorrow against the Czech Republic."
Alain Jardel, Head Coach, France
"I think that Serbia and Montenegro have a better team than us. When you shoot so badly it is impossible to win. We made things compliacated and forgot the basics and this match proved that is the basics that allow you to win."
"The game against Poland is another day, another match."
Cathy Melain, France
"We lost this game but not because of the last moments. It's not over yet and we still have another 3 games in the group."
"We spent a lot of energy getting back into the game against a good team. You have to remember that they were better than us. We tried to get over the first quarter of the game which was a catastrophe."
"To blame the final moments of the match would be too easy of an excuse."