Built in 1905, it is the only survivor of six, three-cylinder cars made by Henry Royce as he was forging his partnership with CS Rolls.
The vehicle started life as a demonstration car before being sold to a man in Yorkshire and then a woman in Dumfriesshire.
It was then sold to Adam Dick, a Kilmarnock garage owner, who called it 'the old girl' and left it in his will to the Royal Scottish Automobile Club in 1974. It went on show in the foyer of its headquarters in Blythswood Square.
RSAC trustee Mike Gascoigne said: "Apart from a short period of initial ownership and when it was in Essex for restoration or on display in the Midlands by the Rolls-Royce Enthusiasts Club, this wonderful old car has always been in Scotland.
"The trustees are delighted at the outcome of our 15-year quest and that the 'old girl' is now an exhibit in Glasgow's stunning Riverside Museum."