The initial cluster mass function (ICMF) in spiral galaxy discs is constrained and compared with data for old globular clusters and young clusters in starbursts. It is found that the observed ages and luminosities of the brightest clusters in spiral discs can be reproduced if the ICMF is a Schechter function with a cut-off mass (Mc) of a few times 105 Msun and disruption of optically visible clusters is dominated by relatively slow secular evolution. A direct Schechter function fit to the combined cluster MF for all spirals in the sample studied here yields Mc = (2.1+/-0.4)x105 Msun. The MFs in cluster-poor and cluster-rich spirals are statistically indistinguishable. An Mc=2.1x105 Msun Schechter function also fits the MF of young clusters in the Large Magellanic Cloud. If the same ICMF applies in the Milky Way, a bound cluster with M>105 Msun will form about once every 10 Myr, while an M>106 Msun cluster will form only once every 50 Gyr. Luminosity functions (LFs) of model cluster populations drawn from an Mc=2.1x105 Msun Schechter ICMF generally agree with LFs observed in spiral galaxies. It is thus concluded that the ICMF in present-day spiral discs can be modelled as a Schechter function with Mc = 200,000 Msun. However, the presence of significant numbers of M>106 Msun (and even M>107 Msun) clusters in some starburst galaxies makes it unlikely that the Mc value derived for spirals is universal. In high-pressure environments, such as those created by complex gas kinematics and feedback in mergers, Mc can shift to higher masses than in quiescent discs.