The Royal Botanic Gardens in Sydney, is the most central of the three major botanical gardens open to the public in Sydney (the others being the Mount Annan Botanic Garden and the Mount Tomah Botanic Garden). The gardens were opened in 1816, and are managed by the same trust that manages the adjoining The Domain. The gardens are open every day of the year, and access is free.
http://en.wikipedia.org/wiki/Royal_Botanic_Gardens,_Sydney