Mr. Pedersen was speaking following a meeting with the government and opposition co-chairs of the Syrian Constitutional Committee, who have agreed to start the process for constitutional reform.
The members of its so-called "small body", tasked with preparing and drafting the Constitution, are in the Swiss city for their sixth round of talks in two years, which begin on Monday.
Their last meeting, held in January, ended without progress, and the UN envoy has been negotiating between the parties on a way forward.
“The two Co-Chairs now agree that we will not only prepare for constitutional reform, but we will prepare and start drafting for constitutional reform," Mr. Pedersen journalists.
"So, the new thing this week is that we will actually be starting a drafting process for constitutional reform in Syria."
The UN continues to support efforts towards a Syrian-owned and led political solution to end more than a decade of war that has killed upwards of 350,000 people and left 13 million in need of humanitarian aid.
An important contribution
The Syrian Constitutional Committee was formed in 2019, comprising 150 men and women, with the Government, the opposition and civil society each nominating 50 people.
This larger group established the 45-member small body, which consists of 15 representatives from each of the three sectors.
For the first time ever, committee co-chairs Ahmad Kuzbari, the Syrian government representative, and Hadi al-Bahra, from the opposition side, met together with Mr. Pedersen on Sunday morning.
He described it as "a substantial and frank discussion on how we are to proceed with the constitutional reform and indeed in detail how we are planning for the week ahead of us."
Mr. Pedersen told journalists that while the Syrian Constitutional Committee is an important contribution to the political process, “the committee in itself will not be able to solve the Syrian crisis, so we need to come together, with serious work, on the Constitutional Committee, but also address the other aspects of the Syrian crisis.”