A study of the three-dimensional axisymmetry-breaking instability of an axisymmetric convective flow associated with crystal growth from bulk of melt is presented. Convection in a vertical cylinder with a parabolic temperature profile on the sidewall is considered as a representative model. The main objective is the calculation of critical parameters corresponding to a transition from the steady axisymmetric to the three-dimensional non-axisymmetric (steady or oscillatory) flow pattern. A parametric study of the dependence of the critical Grashof number Gr(cr) on the Prandtl number 0 less than or equal to Pr less than or equal to 0.05 (characteristic for semiconductor melts) and the aspect ratio of the cylinder 1 less than or equal to A less than or equal to 4 (A = height/radius) is carried out. The stability diagram Gr(cr)(Pr,A) corresponding to the axisymmetric - three-dimensional transition is reported for the first time. The calculations are done using the spectral Galerkin method allowing an effective and accurate three-dimensional stability analysis. Tt is shown that the axisymmetric how in relatively low cylinders tends to be oscillatory unstable, while in tall cylinders the instability sets in due to a steady bifurcation caused by the Rayleigh-Benard mechanism. The calculated neutral curves are non-monotonous and contain hysteresis loops. The strong dependence of the critical Grashof number and the azimuthal periodicity of the resulting three-dimensional flow indicate the importance of a comprehensive parametric stability analysis in different crystal growth configurations. In particular, it is shown that the first instability of the how considered is always three-dimensional.