The man, aged in his 50s and from the York area, was arrested today, North Yorkshire Police said.
Miss Lawrence disappeared six years ago and is still missing.
Her father, Peter Lawrence, is a committed Christian. He told Premier: "It is encouraging to know that following all the media activity over the last three weeks from Claudia's 41st birthday to the 6th anniversary of her being missing, North Yorkshire police continue to be active in seeking answers as to what has happened to Claudia.
"It is to be hoped that the matter can be resolved as soon as possible and I encourage people to continue to come forward with information to the police by phoning them on 101."
Peter Lawrence praying for his daughter
Searches are being carried out and will continue for a number of days, police said.
Detective Superintendent Dai Malyn, of North Yorkshire Police, urged people not to identify the arrested man: "To ensure the investigation and legal process are not compromised or potentially damaged in any way during this critical phase in seeking the truth about Claudia's disappearance, North Yorkshire Police strongly advises the media and members of the public against identifying the man who has been arrested.
"This includes naming or publishing images of the man on traditional media platforms or social networking sites.
"I urge everyone to show restraint and patience while we carry out these very important inquiries."
A 60-year-old man was arrested last year in connection with her disappearance and suspected murder but was later released without charge.
A 47-year-old man remains on bail on suspicion of perverting the course of justice.