The incipient effect of buoyancy on the heat loss from a fine hot wire in two-dimensional steady incompressible flow is considered. The wire is taken to be horizontal, the mainstream is taken to be normal to the wire and to be directed upwards at an acute angle α to the vertical, and the product Nε = NG/R3 is assumed to be small, where N, G and R denote respectively the Nusselt, Grashof and Reynolds numbers (defined, in the usual way, in §§2 and 3). Three parts of the flow are distinguished and discussed in turn. These are (i) the wire's thermal wake, (ii) an outer irrotational flow induced as a small perturbation of the mainstream by the thermal wake, and (iii) an inner flow, near the wire, in which diffusion either dominates or balances convection. It is shown that the change caused by buoyancy in the wire's heat loss is due largely to the irrotational flow induced by the wire's thermal wake. When log Nε is significantly large, the change caused by buoyancy in the wire's heat loss is almost entirely due to the irrotational flow. The increment caused by buoyancy in the Nusselt number is then approximately the same as would be produced, in the absence of buoyancy, if the mainstream speed were increased by a factor 1– 2σ−1Nεlog(Nε) cos α, where σ is the Prandtl number.