The vessel lifted off from the Baikonur Cosmodrome in Kazakhstan at 0714 GMT.
The three-man crew is expected to arrive at the ISS on Friday.
The Canadian is the seventh private individual to make the trip. While previous tourists have emphasised science as a motivator, Mr Laliberte says his inspiration lies elsewhere.
"I'm an artistic person and a creator. I'm not a scientific. I'm not an engineer," he told the BBC this month.
"Life has given me some qualities, some assets and I have built up a team of very creative people around the world.
"With those people I think we'll present something that is originally creative and hopefully will have the result of sensitising people toward the situation of water in the world."
Guy Laliberte and his crew making final preparations for their launch
On 9 October he intends to lead a "poetical social" performance from orbit. The two-hour show will link up with contributions from 14 cities worldwide. The whole endeavour will be streamed on the web.
"I start with the simple idea of reading a poem, which will involve characters like the Sun, the Moon and a drop of water," he explained to the Associated Press this week.
"Those characters will then engage in a discussion, which will take the form of a little poetic story that we will read to the population of Earth."
Celebrities, including former US Vice President Al Gore and Irish rock band U2, are set to make appearances.
The purpose of the poem, written by Yann Martel, author of the Man-Booker prize-winning novel Life of Pi, is intended to emphasise the need for all the world's population to have access to clean water.
Opportunities for tourists to visit the ISS are likely to become extremely limited in future. The expected retirement of the US space shuttles in 2010/11 will mean all Soyuz seats are needed to maintain the resident station crews, which now number six individuals.
All the tourists have bought their tickets from the Russian Federal Space Agency through the marketing company Space Adventures.
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