The 60-year-old, who is thought to be from Northumberland, was watching the Scotland versus England match at Murrayfield when he became unwell around 15 minutes into the game.
He was taken by ambulance to Edinburgh Royal Infirmary where he died.
Hundreds of tributes have been left on social networking sites.
The man was watching the game from the stands with a group of friends when he became ill.
One fan said she saw someone being treated from the South stand before being taken away.
Both the Scottish Rugby Union and Rugby Union have expressed their condolences to the man's family.
A statement from Police Scotland confirmed a 60-year-old man had collapsed at the game at around 5.15pm.
It said: "He was taken to ERI where he was pronounced dead. Inquiries are still ongoing but it is believed there are no suspicious circumstances."
Scottish Rugby Union president Donald Macleod said the Union was "extremely saddened" by the news. He said: "We extend our deepest sympathy to the gentleman's family and friends and we are standing by to support in any way we can."
England won the Six Nations match 20-0.
Rugby Football Union president Bob Reeves said: "The rugby family has lost one of its members and on behalf of all the travelling England fans and the RFU we extend our condolences to the gentleman's relations and all those who knew him."