He accretion on to carbon-oxygen white dwarfs (CO WDs) plays a fundamental role when studying the formation of Type Ia supernovae (SNe Ia). Employing the mesa stellar evolution code, we calculated the long-term evolution of He-accreting CO WDs. Previous studies usually supposed that a WD can grow in mass to the Chandrasekhar limit in the stable He burning region and finally produce an SN Ia. However, in this study, we find that off-centre carbon ignition occurs in the stable He burning region if the accretion rate is above a critical value (similar to 2.05 x 10(-6) M-circle dot yr(-1)), resulting in accretion-induced collapse rather than an SN Ia. If the accretion rate is below the critical value, explosive carbon ignition will eventually happen in the centre producing an SN Ia. Taking into account the possibility of off-centre carbon ignition, we have re-determined the initial parameter space that produces SNe Ia in the He star donor channel, one of the promising channels to produce SNe Ia in young populations. Since this parameter space is smaller than was found in the previous study of Wang et al. (2009), the SN Ia rates are also correspondingly smaller. We also determined the chemical abundance profile of the He-accreting WDs at the moment of explosive carbon ignition, which can be used as initial input for SN Ia explosion models.