Queensferry Crossing still on course to open by end of August
However, there are "too many variables" to be able to be more specific yet, including the weather.
Mr Brown said: "Until we have real confidence about the opening date, we do not want to go public with that."
The bridge was originally due to have opened last December, but it was postponed to May by bad weather hampering construction.
In March, further bad weather put back the May date.
Mr Brown said he had visited the bridge on 19 May and was "hugely impressed" by the progress being made."
He said all traffic would initially use the Queensferry Crossing while the Forth Road Bridge was prepared for buses and taxis.
During that three to four week period, the speed limit on the new bridge would be reduced to 50mph.
Following that, a 70mph limit would be introduced and the Queensferry Crossing will become a motorway.