The diffusion of a passive scalar (heat) from a continuous line source placed in a uniformly sheared, nearly homogeneous, turbulent shear flow is examined. Measurements indicate that the mean temperature profile is approximately Gaussian near the source but, further downstream, it becomes asymmetric and its peak shifts towards the region of lower velocity. The r.m.s. temperature fluctuation profile is double peaked close to the source, it is single peaked at intermediate distances and it demonstrates a re-emergence of double peaks which grow in relative magnitude far away from the source. In comparison to similar experiments in isotropic turbulence, the centreline mean temperature appears to have a comparable decay rate, whereas the centreline mean-square temperature fluctuations appear to decay at a faster rate. The spread of the plume is faster than that in isotropic turbulence. The measured turbulent heat fluxes and triple temperature-velocity correlations demonstrate self-similar features. The development of temperature integral lengthscales and microscales is comparable to that in other heated, uniformly sheared flows, while the temperature p.d.f. and the temperature-velocity joint p.d.f. are distinctly non-Gaussian, especially away from the centreline. The relative magnitudes of the two measured components of the turbulent diffusivity tensor are in agreement with earlier measurements.