Exact solution of the Navier-Stokes equation for a steady vortex in which the diffusion of vorticity is balanced by vortex stretching in an external strain field. The vortex has a Gaussian shape. For example, for a vertical axisymmetric vortex in an external velocity field given by (u,v,w)=gamma (-x/2,-y/2,z), the vorticity is omega = omega_0 exp(-gamma (x^2 + y^2)/4 u) in which u is the kinematic viscosity.