The firm hit the headlines in 2011 when two minor earthquakes in Lancashire were linked to its exploration activity in the region. Operations were suspended pending a government review.
Lambert has joined Cuadrilla in the new role of government and public affairs director, reporting to chief executive Francis Egan. The news comes two months after the Government announced exploration could resume subject to tougher controls.
The Department of Energy and Climate Change said it planned to introduce a new regulatory body, called the Office of Unconventional Gas and Oil.
Cuadrilla is now attempting to secure planning permission and regulatory approvals for further exploration activity in Lancashire.
According to Lambert, Bell Pottinger has been advising the company on government and external relations and will continue to do so. In addition, stakeholder engagement agency PPS is helping Cuadrilla consult with the Lancashire community.
Betfair, the online betting exchange, is not directly replacing Lambert. His duties have been divided between Martin Cruddace, group director of legal and corporate affairs, and Susannah Gill, UK head of public affairs.