Pedestrians tested the bridge's stability
The £18.2m bridge swayed under the weight of thousands of pedestrians when it first opened in June 2000.
After just three days the bridge's operators had to close the 320-metre long structure, which crosses the River Thames between St Paul's Cathedral and the Tate Modern.
A series of shock absorbers has been installed to reduce the movement.
One of the first people to cross the bridge on Friday said: "There are no problems at all, it is completely rigid today."
Engineers wrestled for months with the problem before submitting their solution in September 2000.
In February 2001, the Millennium Bridge Trust announced that it had raised the £5m needed to carry out the necessary modifications to the bridge.
The remedial work started in May 2001.
The modifications involved installing a series of dampers mostly underneath the bridge deck.
The overall design is not compromised.
The work was completed in January this year, after which a series of tests was carried out to ensure that the problem was solved.
The Corporation of London officially took over ownership of the bridge from one 0001 GMT on Friday.
The Millennium Bridge is London's first dedicated pedestrian bridge.
It is also the first new river crossing in central London for more than 100 years, since Tower Bridge was opened in 1894.
Judith Mayhew, chairman of policy and resources at the Corporation of London, said the city was "delighted" by the opening.
"The bridge provides an invaluable and unique link between the communities north and south of the river and we know that it will play a significant role in the regeneration of Bankside, particularly helping the people of Southwark," she said.
Roger Ridsill-Smith project manager said: "We are very proud of this bridge and what we have done.
"As a result of what we have done here footbridges all over the world will be designed in a different way.
"We can predict this phenomenon and design against it."
Shortly before the reopening, the bridge's sculptor, Sir Anthony Caro said: "It was very disappointing when this beautiful bridge opened and then, almost immediately, had to shut again.
"Since then, it's got a trademark and I'm very pleased that after all this time of waiting, it's opening again."
'No wobbles'
Millennium Bridge Trust chairman David Bell said: "Thanks to all that's happened, the structure is probably the best known modern bridge in the world.
"Because it is a suspension bridge, people on it will feel some movement, but tests have shown it wont repeat the wobbles of before.
""This is a fantastic new connection across London.
"Our original estimate was that four million people a year would use the bridge and we think it will be extremely popular."