Rob Stewart was speaking after plans were unveiled for 3,500-seat arena, an aquarium and a city beach area with bars, restaurants, hotels and homes.
It will be the biggest landscape change since the city was blitzed during World War Two and is expected to lead to about 1,700 permanent jobs.
Swansea council unveiled the plans on Friday with work due to start in 2017.
It comes 10 years after a flagship scheme to redevelop Swansea city centre collapsed.
Mr Stewart said: "We've been very careful to make sure what we've proposed is deliverable. We're very confident we have the right developers and right funders."
Two developers - Rivington Land and Acme and Trebor Developments - have been appointed and construction work will bring about 500  jobs.
The main areas to be worked on are the former St David's shopping centre, land next to the leisure centre and the civic centre site.
Mr Stewart added: "People quite rightly would want to see us get on with this as soon as possible.
"We are already in discussions with Welsh government, with the UK department of trade and industry in London, to make sure we can take these plans forward as quickly as possible."
It will run from Whitewalls, include a cafe quarter and be built on an incline so that at the other end, people can walk straight over Oystermouth Road to where the arena will be, hosting concerts, shows, conferences and ice skating.
There will also be a high-rise building next to it comprising of a hotel and flats and could become the tallest in Wales.
An access route will run from there to the aquarium and city beach.
A new £500m plan to transform Swansea city centre will succeed, the leader of the city's council has said.