The council says it is “delighted” by the announcement.
Beckmead was chosen following a rigorous and competitive application and interview process.
Currently, the Trust runs the Beckmead Family of Schools in Croydon, which are rated Outstanding by Ofsted.
It has just taken on an alternative provision academy in Essex, the Beckmead Moundwood Academy.
Richmond’s new school is expected to open in 2023 at the Barnes Hospital site.
It will eventually accommodate up to 90 pupils aged seven to 19.