We give a description of all (1,2)–knots in S3 which admit a closed meridionally incompressible surface of genus 2 in their complement. That is, we give several constructions of (1,2)–knots having a meridionally incompressible surface of genus 2, and then show that any such surface for a (1,2)–knot must come from one of the constructions. As an application, we show explicit examples of tunnel number one knots which are not (1,2)–knots.