She will take up the post in August, succeeding Gordon Seabright who has been acting Director General since September 2009.
Biggs joins the RHS with 30 years experience in the travel and international leisure industry.
After founding Budget Holidays' tour operation in the USA she began a 25 year career at Kuoni.
She became the tourism giant's youngest director, first female and non-Swiss board member and eventually rose to become managing director in 1999.
In 2008 she moved to Thomas Cook to become managing director of Scheduled Businesses.
A longstanding member of the RHS, she has designed six gardens for friends and describes herself as a passionate gardener.
RHS President Giles Coode-Adams said: "I am delighted to welcome Sue to the RHS. She brings a wealth of business experience combined with a real love of plants and gardening, and the vision to help us realise the full potential of this marvellous organisation."
President-elect and chair of the Appointment Committee Elizabeth Banks said: "We were fortunate to have an extremely high calibre of candidates. Sue will be an exceptional leader for the organisation."
Biggs said: "The position of Director General of the RHS is a dream come true for me. I'm looking forward to building on the success of an organisation that I have supported, as a member, for a long time."
Giles Coode-Adams paid tribute to Gordon Seabright's achievements as acting Director General. "I would like to thank Gordon for his role in guiding the organisation through the past nine months. He has brought stability and significant progress after a challenging period and has been a great help to me and to Council. I know that, as part of the senior management team, Gordon will continue to help take the RHS forward."
Seabright will retain his post as commercial director of the RHS