We investigate the (large-scale) bar fraction in a mass-complete sample of M > 10(10.5) M-circle dot 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 (10(10.5) < M < 10(11) M-circle dot) the bar fraction in early-type discs is much higher, at all redshifts, by a factor of similar to 2, than that in late-type discs. This trend is reversed at higher stellar mass (M > 10(11) M-circle dot), 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 10(10.5) < M < 10(11) M-circle dot discs is built up by a factor of similar to 2 over the redshift interval explored, while for M > 10(11) M-circle dot discs it remains roughly constant. This indicates that, already by z similar to 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 similar to 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).