Moscow was selected to host the Championship last month after Tokyo, which was the original host of the event, was forced to pull out due to the devastating earthquake and tsunami that hit Japan on March 11.
"I would like to thank the International Skating Union (ISU) and the Russian Organising Committee of the Championship for their work on preparations for the tournament and I would like to wish all people of Japan courage and firmness in overcoming the aftermath of the natural disaster," said Medvedev.
Russian officials had only a month to get the the stadium, hotels and transport ready for the skaters and spectators.
"This is an unprecedented occurrence that the World Championships was organised [this fast]," Alexander Gorshkov, the President of the Russian Figure Skating Federation.
The Championships, which are due to be held until May 1, are being held at Moscow's Megasport Arena, the city's best ice venue which has hosted for the past three years the Cup of Russia and an ISU Grand Prix of Figure Skating series event.
More than 3,000 athletes, coaches, judges and officials and 500 journalists have travelled to the Russian capital for the event.
Up to 70,000 fans are expected to be attracted to the event and already 80 per cent of tickets have been sold.
According to an initial estimation by Russian State Duma's Sports Committee, it will cost $6 million (£3.7 million) to organise the Championships, which Prime Minister Vladimir Putin has promised that they will pay.
But Moscow Mayor Sergei Sobyanin revealed that about 207 million rubles (£4.5 million/$7.4 million) has already been spent on the Championships.
Besides the budget, the Russian Government has promised "easy access to entry visa and security" for the event participants.
Staff from the Russian Olympic Committee also joined in the organisation, using it as a rehearsal for the 2014 Sochi Winter Olympic Games.
The Championships been held in Russia three times before.
Moscow hosted the event in 2005, and St. Petersburg hosted the very first Championships in 1896 and then again in 1903.
March 2011: Moscow chosen to replace Tokyo as host for World Skating Championships
March 2011: Putin offers to step in and pay for Moscow to host World Skating Championships
March 2011: Earthquake forces cancellation of World Skating Championships
March 2011: World Figure Skating Championships in Tokyo in doubt after earthquake