Huge injury blows for France
Meanwhile, teenage flyhalf Matthieu Jalibert, who made his debut in Saturday's dramatic 13-15 Six Nations defeat, will likely miss the rest of the tournament, likewise due to a knee injury.
Dupont, 20, suffered a ruptured cruciate ligament during a nine-minute cameo on Saturday, while the 19-year-old Jalibert has a partially torn knee ligament.
"He will be out for a minimum of one month, perhaps two," said Brunel of Jalibert.
The Bordeaux-Begles No.10 could be fit again for France's final two tournament matches, at home to England on March 10 and away to Wales a week later.
Back-row forward Kevin Gourdon will be out for up to a month after twisting his ankle against Ireland, meaning he will certainly miss next week's trip to Scotland and possibly also the clash with Italy in Marseille on February 23.
Centre Henry Chavancy has been passed fit, though, after a suspicion of an ankle injury and fractured eyebrow.
The injuries mean there could be a return to the France squad for experienced Toulon flyhalf Francois Trinh-Duc, who was surprisingly left out by Brunel.
Dupont's most likely replacements will be either Lyon's Baptiste Couilloud or Alexi Bales of La Rochelle.
Gourdon's injury could see Louis Picamoles recalled to the squad after poor recent form saw him left out of Brunel's original group.
Former Toulouse and Northampton No.8 Picamoles responded to his France snub with an impressive performance in French Top 14 leaders Montpellier's 30-29 win at champions Clermont last week.
Brunel, who made his debut as France coach against Ireland, already faces the task of rebuilding his team, which has now gone seven Tests without a victory, before next week's trip to Murrayfield.
"It's a slight hammer blow, but the competition continues," Brunel said of the last-gasp Ireland defeat.
"You have to get up, revive the spirits a bit and start tomorrow.
"The tournament does not allow us to dwell on the past. We must try to move forward quickly."
Brunel also admitted that it may have been a mistake to let replacement flyhalf Anthony Belleau, who looks in line to start against Scotland, take a crucial late penalty that could have handed France victory.
Maxime Machenaud had started the match as kicker and was back on the field after Dupont's injury, but Belleau, who had taken over kicking duties once the scrumhalf was substituted, took the shot at goal instead – and missed.
"Maxime had just come back on, Anthony said he felt good," said Brunel.
"Was it a mistake? Yes, maybe. Maybe Maxime could have done what was needed. But he had just returned, perhaps he wasn't in the right frame of mind."