Edwards will head up the escorted touring brand alongside his existing role at the adventure specialist, Explore Worldwide.
The move follows discussions between O'Neill and Ben Boesch, Chief Executive of Dertour UK, about the future direction of the business.
Explaining the move, O'Neill said: "This was not an easy decision, but it feels like the right time for me to begin a new chapter."
Boesch added O'Neill's 36-year contribution to the business had been "extraordinary", praising her "authentic, people-centred leadership" and the impact she had made on colleagues across Dertour UK.
"Her passion for the brand, its people and its customers has helped shape the business we know today," he said. "Debbie leaves with our sincere thanks, appreciation and respect."
He added that Edwards was "well placed to build on the brand's strengths and lead the business through its next phase".
O'Neill joined Jules Verne in 1990 and called her decades-long stint there a "privilege" and a "huge part" of her life. "I am incredibly proud of our people, our culture and everything the team has achieved together," she said. "I remain hugely confident in Jules Verne and excited about its future."
O’Neill will work with Edwards over the coming weeks to ensure a smooth handover, while looking ahead to spending more time on "developing people, supporting leaders and helping individuals and teams fulfil their potential".
This people-focused approach has also been central to O'Neill’s work beyond Jules Verne. In an interview with TTG in August, in her role as chair of the Association of Women Travel Executives, she called for greater male allyship in tackling sexism in the travel industry.
"We need the men to stand alongside us and do the work with us," she told TTG at the time.