Although the critical Reynolds number for linear instability of the laminar flow in a straight pipe is infinite, we show that it is finite for a divergent pipe, and approaches infinity as the inverse of the divergence angle. The velocity profile at the threshold of inviscid stability is obtained. A non-parallel analysis yields linear instability at surprisingly low Reynolds numbers, of about 150 for a divergence of 3°, which would suggest a role for such instabilities in the transition to turbulence. A multigrid Poisson equation solver is employed for the basic flow, and an extended eigenvalue method for the partial differential equations describing the stability. © 2005 Cambridge University Press.