Goddard has overseen the restoration of the Victorian Garden at the Cathedral of St John the Baptist in Norwich and most recently she was manager of Painshill Park, Surrey.
She has a background in theatre, the arts and cultural events development, whereas Bopp came from a horticultural background.
Hampshire County Council leader Councillor Roy Perry said: "Clare has a fantastically diverse career in the heritage, business and arts sectors and we are sure that under her leadership, the gardens will continue to grow in popularity with visitors and members of all ages through the fantastic work being done to promote horticulture, conservation, education and recreation."
Goddard said: "I am overjoyed, and consider it a great privilege, to have been appointed director of the Sir Harold Hillier Gardens. I very much look forward to working alongside all the talented staff, volunteers, supporters and public who make it the very special place that it is today. It is an honour to be joining such a prestigious organisation at this exciting stage in its development."
The 180 acre gardens have 12,000 species of plants, 14 national plant collections, 400 champion trees and attract more than 200,000 visitors annually, with 22,000 members.
Sir Harold Hillier left the gardens to the council in 1977.