"I grew up on the South Bank, which is why this theatre is so special to me," said Law, stretching the definition of South Bank to include Lewisham.
"The South Bank is where theatre really started in this country. It is where people came to see the plays of Shakespeare and Marlowe, and it seems apt that the ghosts of the past should be housed here.
"The Young Vic has been here for 34 years and anything we can do to make it go on another 34, 64 or 94 years is important."
Despite claims from the theatre that Law was going to "brave the paparazzi", he spent just 30 minutes in the building on Tuesday for a photo session with two selected journalists whilst other photographers were left on the pavement. Law was travelling from Rome to Paris via The Cut, and had to be whisked to Waterloo station in a Mercedes to catch the 12.09 Eurostar.
Built in 1970 as a temporary structure, the breeze block building in The Cut is now in a serious state of disrepair.
Of the £12.5 million redevelopment cost, £5 million has already been raised through donations and grants, and the same sum is expected in lottery money later this month.
That leaves a further £2.5 million needed from the public – some of which will be met by Law himself, who is the theatre's patron.
Law, 31, has since appeared on stage there twice, most recently in a production of Marlowe's Doctor Faustus in 2002.
The theatre's artistic director David Lan said: "The building is now genuinely falling down. We are rebuilding because if we don't, the company will close.
"We know we have the public's support. We are asking them to show it now, in big ways and small. Every penny helps. Every pound helps 100 times more."
The public will be able to donate money to the campaign in a variety of ways. They will be able to buy a seat in the new auditorium and have it named after them for £1,000; contribute an extra £1 with every ticket bought; or make a dedicated gift towards a particular room in the theatre, from a toilet at £2,500 to a dressing room for £50,000 or the main auditorium for £1 million.
The rebuilding campaign has been running for several years. In September 2001, Tony and Cherie Blair attended a fundraising gala in aid of the Waterloo theatre. The scheme got another big boost in September 2003 when Patrick McKenna donated £1 million at a gala attended by Zoe Wanamaker, Joseph Fiennes, Stephen Daldry, Greg Dyke, Corin Redgrave and Philip Pullman.