Spain international Ramos, 28, suffered a torn hamstring in Wednesday's 2-1 La Liga win over Sevilla and limped off after just nine minutes before being replaced by Nacho.
"Sergio's injury is a serious one. He'll be out for five weeks," Ancelotti explained.
Ramos, who had already been ruled out of Saturday's derby against Atlético, could also miss both legs of Champions League round of 16 against Schalke, on 18th February and 10th March.
The centre-back's absence is a major blow for Real, who already had to do without James Rodriguez (foot), Pepe (rib) or Luka Modric (thigh).
Against their Madrid rivals, Real are set to start with Raphael Varane, 21, and Nacho, 23, at the heart of the defence.
"I trust those two players. They are very good players. They are dedicated and willing to play. I am convinced they will do good things on Saturday", Ancelotti said.
Real, currently top of the league with 54 points, face Atlético on Saturday for the tenth time in the past year in all competitions.