We study the effect of a central mass concentration (CMC) on the secular evolution of a barred disc galaxy. Unlike previous studies, we use fully self-consistent 3D N-body simulations with live haloes, which are known to be important for bar evolution. The CMC is introduced gradually, to avoid transients. In all cases for which the mass of the CMC is of the order of, or more than, a few per cent of the mass of the disc, the strength of the bar decreases noticeably. The amount of this decrease depends strongly on the bar type. For a given CMC, bars with an exponential surface-density profile, which formed in a disc-dominated galaxy (MD-type bars), can be totally destroyed, while strong bars with a flat surface-density profile, whose evolution is significantly affected by the halo (MH-type bars), witness only a decrease of their strength. This decrease occurs simultaneously from both the innermost and outermost parts of the bar. The CMC has a stronger effect on the Fourier components of higher azimuthal wavenumber m, leading to fatter and/or less rectangular bars. Furthermore, the CMC changes the side-on outline from peanut-shaped to boxy or, for massive CMCs, to elliptical. Similarly, side-on initially boxy outlines can be destroyed. The CMC also influences the velocity dispersion profiles. Most of the decrease of the bar strength occurs while the mass of the CMC increases, and it is accompanied by an increase of the pattern speed. In all our simulations, the mass of the CMC necessary in order to destroy the bar is at least several per cent of the mass of the disc. This argues that observed supermassive black holes are not likely to destroy pre-existing bars.