We investigate the (large-scale) bar fraction in a mass-complete sample of M > 1010.5 M⊙ disc galaxies at 0.2 < z < 0.6 in the Cosmological Evolution Survey (COSMOS) field. The fraction of barred discs strongly depends on mass, disc morphology and specific star formation rate (SSFR). At intermediate stellar mass (1010.5 < M < 1011 M⊙) the bar fraction in early-type discs is much higher, at all redshifts, by a factor of ∼2, than that in late-type discs. This trend is reversed at higher stellar mass (M > 1011 M⊙), where the fraction of bars in early-type discs becomes significantly lower, at all redshifts, than that in late-type discs. The bar fractions for galaxies with low and high SSFRs closely follow those of the morphologically selected early- and late-type populations, respectively. This indicates a close correspondence between morphology and SSFR in disc galaxies at these earlier epochs. Interestingly, the total bar fraction in 1010.5 < M < 1011 M⊙ discs is built up by a factor of ∼2 over the redshift interval explored, while for M > 1011 M⊙ discs it remains roughly constant. This indicates that, already by z∼ 0.6, spectral and morphological transformations in the most massive disc galaxies have largely converged to the familiar Hubble sequence that we observe in the local Universe, while for intermediate-mass discs this convergence is ongoing until at least z∼ 0.2. Moreover, these results highlight the importance of employing mass-limited samples for quantifying the evolution of barred galaxies. Finally, the evolution of the barred galaxy populations investigated does not depend on the large-scale environmental density (at least, on the scales which can be probed with the available photometric redshifts).