The use of iris loaded cavities to accelerate high current bunched beams makes it important to understand the impedance and wake fields caused by an iris in a beam pipe. In this paper we present a method to calculate both the longitudinal and transverse impedance of a circular iris of radius b and thickness g in a circular beam pipe of radius a for ultrarelativistic particles. An integral equation is derived for the transverse electric field at the junction between the iris and the beam pipe and a variational expression is obtained for the impedance, using the transverse field as a trial function. Accurate numerical results are obtained for the longitudinal and transverse impedances using a trial function with only a few adjustable parameters. By invoking causality we confirm the analytic behavior of the impedances in the complex frequency plane and obtain the corresponding wake functions. We particularly explore the limit b→∞ to compare with previous studies of the impedance of a circular hole in a transverse metallic plane.