Wales are due to visit Murrayfield on February 13.
Scotland are also due to play home matches against Ireland and Italy, on March 14 and 20 respectively. The SRU said the possibility of having fans at those games would be “reviewed in the weeks ahead”.
The letter said: “We are writing to confirm that, to comply with the current Scottish Government public health guidance in relation to COVID-19, Scotland’s opening Guinness Six Nation match against Wales at BT Murrayfield on Saturday 13 February, will be played behind closed doors.
“Scottish Rugby is continuing to work closely with the Scottish Government in supporting efforts to tackle this virus, and access to Scotland’s two other home Guinness Six Nations games, against Ireland and Italy, will be reviewed in the weeks ahead. We will email all Debenture holders as soon as we know more about these games.
"This latest news is naturally very disappointing, and everyone at Scottish Rugby very much looks forward to welcoming you back to BT Murrayfield.”
Speaking in December, SRU chief executive Mark Dodson hadn’t given up on fans attending.
“We are still hopeful we can get crowds into the Six Nations,” he said at the SRU agm. “Two of our three homes games are at the end of the Six Nations at the end of March so we’ve got more chance with them but it’s something we have to talk to government about.”