This paper presents an analytical treatment for the unsteady one-dimensional natural convective flow past an infinite moving vertical cylinder in the presence of thermal stratification. Exact solutions of the dimensionless unsteady coupled linear governing equations are obtained, in terms of Bessel functions by the Laplace transform technique, for the tractable case of unit Prandtl number. Numerical computations for velocity, temperature, skin-friction, and Nusselt number are made for various set of physical parameters and presented in graphs. Due to the presence of thermal stratification, the fluid velocity and temperature approach steady state, whereas the corresponding flow in an unstratified fluid does not. The steady state is attained at smaller times as the stratification increases. Furthermore, in the presence of stratification, the skin-friction and Nusselt number approaches fixed value as time progresses, while for unstratified fluid, there is a gradual decrease as time increases.