Brighton and Hove Buses said that the crash on Monday was an unforeseeable accident but that it had accepted responsibility.
And the company also promised to support the passengers who were involved in the crash in North Street, close to the Clock Tower.
The driver, Mark Gould, from Southwick, remains in a serious condition in the Royal Sussex County Hospital, in Brighton.
Managing director Martin Harris said: “As we come to the end of the week in which the terrible accident in North Street occurred, I would like to wish all the passengers who were injured, and our own driver, the most successful and speedy of recoveries.
“The accident on Monday morning was exactly that. A terrible accident.
“The medical episode that affected our driver minutes before the collision could not have been foreseen by him. It could not have been foreseen by us as his employers.
“There was no previous history of this condition. There were no symptoms of this condition apparent when he declared himself fit to work.
“Nevertheless, I can confirm that Brighton and Hove Bus Company accepts responsibility for the incident and we will continue to work with those investigating the incident and will support those passengers involved.
“Thank you to everyone who has sent in cards and wishes for our driver and for everybody at the company.
“I am extremely proud of the work that our staff do in difficult circumstances and they have rallied round brilliantly this week.
“We would all like to publicly state our gratitude to the emergency services for their outstanding work throughout this incident once again.”