We report on a 120-ks XMM-Newton observation of the galaxy cluster Abell 2597 (A2597). Result. from both the European Photon Imaging Camera (EPIC) and the Reflection Grating Spectrometer (RGS) are presented. From EPIC we obtain radial profiles of temperature, density and abundance, and use these to derive cooling time and entropy. We illustrate corrections to these profiles for projection and point spread function (PSF) effects. At the spatial resolution available to XMM-Newton, the temperature declines by around a factor of 2 in the central 150kpc or so in radius, and the abundance increases from about one-fifth to over one-half solar. The cooling time is less than 10Gyr inside a radius of 130kpc. EPIC fits to the central region are consistent with a cooling flow of around 100 solar masses per year. Broad-band fits to the RGS spectra extracted from the central 2arcmin are also consistent with a cooling flow of the same magnitude; with a preferred low-temperature cut-off of essentially zero. The data appear to suggest (albeit at low significance levels below formal detection limits) the presence of the important thermometer lines from Fexvii at 15-17 Årest wavelength, characteristic of gas at temperatures ∼0.3keV. The measured flux in each line is converted to a mass-deposition estimate by comparison with a classical cooling flow model, and once again values at the level of 100 solar masses per year are obtained. These mass-deposition rates, whilst lower than those of previous generations of X-ray observatories, are consistent with those obtained from ultraviolet data for this object. This raises the possibility of a classical cooling flow, at the level of around 100 solar masses per year, cooling from 4keV by more than two orders of magnitude in temperature.