Summary: We analyze the null controllability of the one-dimensional linear heat equation \(\rho(x)y_t-(a(x)y_x)x=0\) with non-smooth time-independent coefficients \(\rho\) and \(a\). More precisely, we prove that this equation with BV coefficients is null-controllable at any positive time in the context of boundary control. The argument used in the proof relies on the exact controllability of the one-dimensional wave equation with BV coefficients and Russell’s general method (which provides the null controllability of a parabolic equation at any positive time as a consequence of the exact controllability, for large time, of the corresponding wave equation).