The inspirational Robben limped out of last week's 1-0 victory over Blackburn Rovers, after scoring the first-half winner. Jose Mourinho's side failed to build on that lead and drew 0-0 against Manchester City on Sunday. Robben will also miss the Champions' League tie against Barcelona and the Carling Cup final against Liverpool at the end of this month.
Robben said: "This is a disaster. Everything was going so well. It is a massive blow and comes just at the moment when I was in top shape. Six weeks is the average time to heal this injury and then I need a few more weeks to be completely fit."
Mourinho conceded after their goalless draw against Manchester City that his side would miss the penetration offered by Robben's pace.
The 21-year-old missed the first two months of the season with a broken bone in his other ankle and Robben knows he can expect a long time on the treatment table as the season draws to a close. "I had the feeling directly that it was a serious injury but because of the swelling it was impossible to make a final diagnosis," he added.
"It felt different than last summer when I had the same injury on my other foot. Then I could walk already after three days but I stayed sidelined for a long period. I hope it will take me now six to eight weeks."
The Netherlands coach Marco Van Basten has warned Robben that he must learn to tackle differently if he is to stay clear of the foot injuries which have taken the shine off his successful first season with Chelsea.
Van Basten, whose career was ended by an ankle injury at the age of 29, said: "Every defender knows they must eliminate Robben. Maybe he must learn to go differently, more light-footed into a duel. I had to do that in the past. That is something he'll learn."
Chelsea struggled without Robben against City and fear the injury could keep him out of their run-in to the season as they chase four trophies.