She replaces Keith Cooper, who achieved brief national fame in the television documentary series The House, but has now been moved internally to take charge of sales and broadcasting.
Before joining the BBC Ms Grahame was in charge of marketing at the London Philharmonic Orchestra and subsequently The Philharmonia.
The Royal Opera House is currently the subject of a House of Commons select committee inquiry and a separate committee set up by Culture Secretary Chris Smith to investigate options including the Royal Opera, Royal Ballet and English National Opera sharing Covent Garden. It has also come close to liquidation with mounting debts this year.
Ms Grahame, who takes up her appointment in January, will be responsible for presenting the House's public image and liaising with the Government and the media. Her role will encompass external relations for the Royal Opera, Royal Ballet and the Royal Opera House. She said last night: "This must be one of the most challenging jobs in the arts."
The opera house will be thrust into the spotlight again next Wednesday when the select committee issues its report. It is expected to be highly critical of arrangements during the two-year closure of Covent Garden. Opera and ballet companies have had to perform in several different venues, and ticket sales have been low.