The blast was in the Dehmazang area in the west of the city. Kabul police spokesman Mohammad Ismail Kawusi said that in addition to the person killed, 11 had been taken to hospital.
The city was hit by a series of suicide attacks last month, including one that killed seven journalists from a private television station as the Taliban stepped up their campaign against the Afghan government.
The attacks coincided with renewed efforts to revive a peace process with the Taliban that stalled last year.
The Afghan National Civil Order Police (ANCOP) was set up as a gendarmerie-style unit to control riots and urban disorder but have also been used in counterinsurgency roles against the Taliban.