Ms Mason has had a long association with the company since becoming its youngest dancer when she joined aged 16 in 1958.
She has been assistant director for 11 years, and has been acting director since Mr Stretton stepped down in September.
Ms Mason will be seen as a safe choice who is known and liked by those within the company.
During Mr Stretton's tenure, there were complaints from dancers about changes to schedules and uncertainty about who would appear in which performances.
Ms Mason was quick to "restore calm" within the company after Mr Stretton left, according to Royal Opera House chairman Sir Colin Southgate.
Tony Hall, executive director of the Royal Opera House, said Ms Mason made an "enormous contribution" to the company as a dancer, coach and custodian of the Kenneth MacMillan repertory.
"With such extensive knowledge and experience of the company, she is perfectly placed to take the Royal Ballet through the next period of its history until the summer of 2007," he said.
Ms Mason said she was "deeply honoured to be appointed director of this great company".
"Over the last three months I have been overwhelmed by the support given to me by everybody at the Royal Opera House and the dance community at large," she said.
"I am very excited at the prospect of leading the company over the next four-and-a-half years, during which we will celebrate the company's 75th birthday."
Life with the ballet
After joining the company in 1958, Ms Mason moved through the ranks, becoming a soloist and later a principal.
Legendary choreographer Kenneth MacMillan created a number of roles for her, and she was appointed McMillan's répétiteur - or coach - in1980.
Four years later, she became principal répétiteur to the company, and was made assistant director in 1991.
Her appointment as director comes 12 months after she was made an OBE for services to dance.