The former Soviet republic was told to speed up preparations earlier this year by European football's governing body and Infantino toured stadiums and airport terminals in Kiev and other host cities to check on progress.
"There is still a lot to do, but a lot of people are working to ensure that this work will be done," he told a news conference.
"I have full confidence and trust and I'm very optimistic on the success of the Euro 2012 in Ukraine."
The Kiev government alarmed UEFA officials this year by admitting that the country, which is to co-host the 16-team tournament with Poland, was well behind schedule in building stadiums, hotels and roads, and developing airports.
Ukrainian President Viktor Yanukovich said his country had made up a lot of lost ground.
"Our mood has been confirmed by the way we have caught up after being behind on schedule," he said in a letter to UEFA President Michel Platini, published by local media.
The government, which has paid out millions of dollars to speed up construction, says preparations are now back on track.
"We have no problems. Everything will be fine," Borys Kolesnikov, Ukraine's Deputy Prime Minister for Euro 2012, told Reuters on Thursday.
Apart from Kiev, Euro 2012 matches in Ukraine are to be held in Lviv, Kharkiv and Donetsk.
