The vehicle is believed to be the oldest surviving unrestored Mini and will probably be snapped up by a fan of the make wanting to return it to its former glory.
Auctioneers Bonhams say the car was the eighth of its type to roll off the production line at Longbridge in Birmingham in May 1959. It is known to have been owned by one Gladys Hobro of Aldwick, near Bognor Regis, West Sussex, before being bought by David Gallimore in 1986. Gallimore kept it in his garage at Chichester and there are still only 30,041 miles on the clock.
John Polson, of Bonhams, said: "This is a wonderful opportunity to buy the car and restore it. Collectors love the fact it has had very little done to it since it was built.
"The car was made very, very simply without many instruments or equipment.
"It was designed to be an affordable, family car. The Mini is one of the most important cars of the 20th century. They have always been collectable.
"Some collectors would want to return her to new, but others would just like to get her going again and keep her in the original condition."
It is thought that only three Minis earlier than this one exist, but they have been restored. One is owned by the British Motor Industry Heritage Trust and the other two are in private collections in Japan.
The Mini is expected to fetch up to £15,000 when it is auctioned on 30 April in Hendon, north London.