Police said Wednesday that the stabbings took place in various spots in the town of Sarpsborg, some 80 kilometers (50 miles) south of Oslo.
In addition to leaving one woman dead, another was seriously injured. The third woman suffered an injury to her arm.
Police said one of the women had recognised the suspected stabber and that the assailant was later detained in an apartment in central Sarpsborg. It was unclear whether the other two woman knew him.
There are currently no signs of other attackers being involved.
Norwegian state broadcaster NRK said the first stabbing took place at a bus station in central Sarpsborg.
The motive for the attack remains unclear but local newspaper Sarpsborg Arbeiderblad said the suspected attacker had a criminal background with a history of violence. (AP) AMS AMS