Recently, the club confirmed Tweddle would lead the side in an interim head coach role.
This is due to Jason Kerr linking up with Welsh Fire in The Hundred during this period.
Tweddle said: “It’s an absolute honour and I’m very proud to take on the position in Jason’s absence. I love this group of players, I love the Club and I’m excited about the 50-over competition.
“I’m so grateful to those at the Club who are continuing to support me on this journey and providing me with these opportunities.”
Director of cricket, Andy Hurry added: “As I have stated previously, Paul is extremely highly regarded by the players, the coaching staff and across the game.
“This decision has been universally welcomed by the dressing room, and we are looking forward to seeing his ever-evolving leadership skills continuing to build on the growth and learnings that the squad have experienced over the last two seasons within the competition.
“We are excited to see how far we can progress during this season’s One-Day Cup campaign under his stewardship.”
Somerset start their One-Day Cup campaign at home against Warwickshire on Friday, August 4 at 2pm.