Heat transfer from a cylinder placed symmetrically in a constant shear field is considered experimentally for low values of the shear Reynolds number, but for Péclet numbers Pe as large as 2000. With the cylinder held stationary, the experimentally obtained Nusselt number Nu is found to be in excellent agreement with the theoretically derived asymptotic expression, $Nu = 1.641 Pe^{\frac{1}{3}} (Pe \rightarrow \infty)$, for Péclet numbers in excess of 100. In contrast, with the cylinder rotating at a speed corresponding to zero torque, the Nusselt number becomes effectively independent of the Péclet number for Pe > 70. This surprising behaviour, predicted theoretically by Frankel & Acrivos (1968), results from the presence of a region consisting entirely of closed streamlines which surrounds the freely rotating cylinder.