The Legco Secretariat said the driver was last on duty on Friday and always wore a mask at work.
Leung tested negative in a rapid test taken after he learnt of his driver’s suspected infection.
But Leung has decided that he won’t return to the Legco complex until he obtains a negative result for a deep throat saliva test.
Meanwhile, two Legco Secretariat staff members have also caught Covid.
One last worked on February 11 and the other on February 15, and both spent time in meeting rooms.
Lawmakers Peter Shiu, Jimmy Ng and Frankie Yick, who chaired video conferences from the meeting rooms, have been notified of the infections.
Staff members who had recently been in contact with the infected workers have been told to undergo tests, and they can only return to the Legco building after getting negative results back.
The Legco president’s car and various places in the building will be disinfected.