Merkel fell while skiing over the Christmas vacation. What she first thought was heavy bruising turned out to be a partial fracture, meaning she must take it easy for three weeks and work from home where possible, said her spokesman Steffen Seibert.
The news comes a week after Germany was shocked by the far more serious skiing accident of former Formula One champion Michael Schumacher, who is in critical condition in a French hospital.
"The chancellor is of course able to work and is in full communication," said Seibert. He added that Merkel's accident occurred "at low speed" but he declined to give further details.
The 59-year-old chancellor, a keen hiker who has said she prefers cross-country to downhill skiing, has postponed a visit to Warsaw scheduled for Wednesday and a meeting in Berlin with Luxembourg's new prime minister, Xavier Bettel.
But she will lead her first cabinet meeting of 2014 on Wednesday. Merkel began her third term last month after sealing a "grand coalition" of her conservatives and the centre-left Social Democrats (SPD).
Merkel will not attend the World Economic Forum later this month but not because of her skiing accident, Seibert said, adding that the meeting in Davos at the end of January clashed with a scheduled German government meeting.