The couple will tie the knot in a civil ceremony at the palace on July 8 and then hold a religious service the following day, a statement said. The venue of the church ceremony will be announced at a later date.
Prince Albert has ruled Monaco, a city state on the south coast of France and a millionaires' playground, since succeeding his late father in July 2005.
Wittstock is a former swimming champion and schoolteacher who met the prince in 2000 after she won gold in the 200 metre backstroke at a Monaco swimming meeting he was hosting.
It is not clear exactly when the pair became an item, but they appeared together socially at the 2006 Winter Olympics in Turin, at the Monaco Grand Prix and at this year's Swedish royal marriage on June 19.
Their formal engagement was announced four days later.
"This is a marriage of considerable importance," said Stephane Bern, the organiser of Monaco's annual high-society ball and expert on princely affairs.
"You can expect at least 2,000 journalists. Monaco inspires fascination in people's minds, it's a permanent dream factory," he added, suggesting the frenzy could equal that surrounding the wedding of Prince Albert's father.
Rainier's marriage to Kelly in 1956, in which her Hollywood star power met his European aristocratic pomp, was one of the society events of the century and royal-watchers are keenly awaiting Prince Albert's nuptials.
Soon to lose his status as one of Europe's most eligible bachelors, Albert has a reputation as a playboy prince, and Monagesque constitutionalists will hope the marriage produces a royal heir.
With a fortune estimated at some two billion euros (2.4 billion dollars), the prince has been seen with a string of beautiful women over the years.
But, while he has fathered two children -- a girl and a boy who were officially recognised after he had acceded to the throne -- neither can succeed him as Monaco's laws require its rulers to be born in wedlock.
The children are Jazmin Grace, 20, whose mother Tamara Rotolo is a former American waitress, and Alexandre Coste, six, the daughter of a French-Togolese former flight attendant, Nicole Coste.
Given the constitutional urgency, some have wondered why it took so long for Albert to confirm the engagement.
Last year, Bern said that Wittstock was taking the time to learn French and the traditions of the Monagesque court.
Monaco, a super-rich enclave on the Riviera coast surrounded on its landward side by France, covers only 200 hectares (494 acres). It is home to 32,000 permanent residents, only 8,000 of them full citizens.
Low tax rates, a luxury yachting marina and a famed casino have attracted many wealthy expatriates, and around 36,000 mainly French non-residents arrive every day to work in its tourism and financial services businesses.
Prince Albert rules as head of state, supported by a senior French civil servant and a government of four senior advisers.
Born in Bulawayo, Zimbabwe, Wittstock competed for South Africa in the Olympics in Sydney 2000. She did not compete in Beijing in 2008, however, despite pressure from former president Nelson Mandela for her to do so.
Prince Albert is a fellow Olympian, having competed in five Olympic games as a member of his country's bobsleigh team.