In this paper we consider the influence of centrifugal arid Coriolis accelerations on the form and stability of a flame propagating in a rotating tube. We detect a discrete sequence of angular velocities, near which the flame propagation velocity undergoes an unlimited amplification. We show that sufficiently rapid rotation may suppress cellular instability of the flame front. We construct a self-similar solution that describes a flame propagating in a rotating gas in free space. Our predictions of the flame shape and of the suppression of cellular instability in the presence of rotation have been verified in a simple experiment we have performed.