We define a class of real numbers that has full measure and is contained in the set of Roth numbers. We prove the $C^1$ - part of Herman's theorem: if f is a $C^3$ diffeomorphism of the circle to itself with a rotation number ω in this class, then f is $C^1$ --conjugate to a rotation by ω. As a result of restricting the class of admissible rotation numbers, our proof is substantially shorter than Yoccoz' proof.