Police spokeswoman Elizabeth Kanjela was quoted as saying the death toll could rise as soldiers and firefighters were still looking for trapped bodies.
The accident occurred about 100km (60 miles) north of the capital Lusaka on a two-lane highway known for heavy traffic.
Poorly maintained and overloaded vehicles are commonplace in Zambia.
Nonetheless, this is said to be one of the worst traffic accidents in the country’s history.
According to reports, the bus was operated by the Zambia Postal Service, which provides passenger services on buses which carry post, and was full of people.
There was serious speeding by all the vehicles involved, Harry Kalaba – an official in the vice-president’s office – told AFP news agency from the crash site.
Some 22 dazed or unconscious survivors were being treated in a nearby missionary hospital, AFP said, while more seriously injured casualties had been taken to hospital in Lusaka.