Ms Heywood said: "The deal secures the company's future in London while we continue our search for a long-term home for the RSC in the capital.
"We're well advanced now in our plans for a large-scale auditorium in Stratford that Shakespeare would recognise as a theatre. The challenge on the horizon is to find a complementary theatre in London."
Today's news means the company is no longer exposed to attack - and headline-grabbing suggestions by the London Assembly its government funding be stripped - for failing to give Londoners access to its work.
Sir Cameron, whose hugely lucrative musical Les Miserables has run for 20 years and was originally a collaboration with the RSC, said: "I could not be more delighted that the Royal Shakespeare Company has decided to make a five-year commitment to do its London seasons in one of my theatres.
"It has always been my ambition to have an ongoing relationship with an internationally acclaimed subsidised company as part of our programming."
Public booking for the RSC's comedies season in London opens on 1 October. Box office: 0870 609 1110.