The One Show is on BBC One at 7pm each weekday evening and is described as 'If it's got Britain talking then it will get talked about on The One Show.' David's episode was presented by Matt Baker and Angela Scanlon. The show was broadcast live from The One Show studio in W1A.
This page has photos of David on the show.
As well as discussing Don Juan In Soho (with two rehearsal photos shown), there was a section where David and Adrian were behind a green screen with characters projected around them and they had to guess who they were. There was also a section about when food and drink is not edible.
David Tennant on The One Show section of DavidTennantOnTwitter.com