An asymptotic theory describes the stationary instability of convective flow between differentially heated vertical planes at large Prandtl numbers. The theory is concerned with the structure for A [Gt ] 1 where A is a Rayleigh number based on the horizontal temperature difference and the distance between the planes. As such it is relevant to the instability of flow in a vertical slot of aspect ratio h [Gt ] 1 where the convective regime corresponds to order-one values of a non-dimensional parameter γ which partly depends on the vertical temperature gradient generated in the slot and can be approximated by γ4 = A/8h. Instability is shown to set in at a critical value of γ that compares well with experimental observation. The lower branch of the neutral curve conforms to a boundary-layer type approximation while the upper branch has a critical-layer structure midway between the planes which becomes fully developed as the first reversal of the vertical velocity of the base flow is encountered near the centreline.