He will join the board on November 1, replacing David Jones who has served after eight years as a Tigers director.
Sir Digby is a Leicester season ticket-holder.
"Leicester is a club with a great heritage and tradition, and I am honoured to accept this invitation," he said.
Tigers chairman Peter Tom added: "I am delighted to welcome Sir Digby to the board.
"He brings with him a wealth of business knowledge and commercial expertise, which will be invaluable to us as we look to further strengthen our position as the leading rugby club in Europe and embark on an ambitious stadium development programme."