A seamless steel tube has a particular composition and a structure with an average prior γ grain size of 150 μm or less in an axial cross-section after cold drawing and heat treatment. The structure retards the growth of a fatigue crack. The steel tube has a tensile strength TS of 500 MPa or more and good internal pressure fatigue resistance and is suitable for use as a fuel injection tube under high injection pressures. The composition of the steel tube may further contain at least one of Cu, Ni, Cr, Mo, and B; at least one of Ti, Nb, and V; and/or Ca.