Context. Star clusters are present in almost all types of galaxies. Here we investigate the star cluster population in the low-luminosity, unperturbed spiral galaxy NGC 45, which is located in the nearby Sculptor group. Both the old (globular) and young star-cluster populations are studied. Aims. Previous ground-based observations have suggested that NGC 45 has few if any “massive” young star clusters. We aim to study the population of lower-mass “open” star clusters and also identify old globular clusters that could not be distinguished from foreground stars in the ground-based data. Methods. Star clusters were identified using UBVI imaging from the Advanced Camera for Surveys (ACS) and the Wide Field Planetary Camera 2 (WFPC2) on board the Hubble Space Telescope. From broad band colors and comparison with simple stellar population (SSP) models assuming a fixed metallicity, we derived the age, mass, and extinction. We also measured the radius for each star cluster candidate. Results. We identified 28 young star cluster candidates. While the exact values of age, mass, and extinction depend somewhat on the choice of SSP models, we find no young clusters with masses higher than a few 1000 M for any model choice. We derive the luminosity function of young star clusters and find a slope of α = −1.94 ± 0.28. We also identified 19 old globular clusters, which appear to have a mass distribution that is roughly consistent with what is observed in other globular cluster systems. Applying corrections for spatial incompleteness, we estimate a specific frequency of globular clusters of S N = 1.4–1.9, which is significantly higher than observed for other late-type galaxies (e.g. SMC, LMC, M33). Most of these globular clusters appear to belong to a metal-poor population, although they coincide spatially with the location of the bulge of NGC 45.