Any trips made during that time will incur an extra £25 charge each time. Driver’s cars will be fitted with a telematic system, containing a GPS unit, which will track when and where the car is being driven.
The insurance company is offering the deal to reduce the risk of accidents, which are twice as likely to happen at night.
Bryan Turner, head of DriveTime at More Than, said: "Young drivers have often found it difficult to get affordable insurance cover as many companies categorise them all as being high risk.
"We recognised that not all young drivers are the same and wanted to come up with a way to reward safer young motorists with cheaper premiums."
More Than has already started to recruit 2,000 volunteers between 18 and 25 years old as part of a trial of the new technology.