The new grandstand, with capacity for 6,500 spectators, will replace a smaller 1920s block where private boxes including the one for the royals are located.
The Jockey Club, which has owned and operated the track in western England since 1964, said the investment was its biggest in any of its racecourses. The existing grandstand will be modernised as part of the work, scheduled for completion by 2016.
Held in March every year, the four-day Cheltenham Festival is one of the highlights of the jump racing season. It attracted aggregate crowds of more than 235,000 earlier this month.
Subject to planning permission, work will begin after next year's festival and the course will remain open for its normal racing programme during the improvements.