We report the results of a 2-month campaign conducted with the Chandra X-ray observatory to monitor the ultraluminous X-ray source (ULX) NGC 5204 X-1. This was composed of a 50-ks observation, followed by ten 5-ks follow-ups spaced initially at ~ 3, then at ~ 10 day intervals. The ULX flux is seen to vary by factors ~ 5 on timescales of a few days, but no strong variability is seen on timescales shorter than an hour. There is no evidence for a periodic signal in the X-ray data. An examination of the X-ray colour variations over the period of the campaign shows the ULX emission consistently becomes spectrally harder as its flux increases. The X-ray spectrum from the 50-ks observation can be fitted by a number of disparate spectral models, all of which describe a smooth continuum with, unusually for a ULX, a broad emission feature evident at 0.96 keV. The spectral variations, both within the 50-ks observation and over the course of the whole campaign, can then be explained solely by variations in the continuum component. In the context of an optically-thick corona model (as found in other recent results for ULXs) the spectral variations can be explained by the heating of the corona as the luminosity of the ULX increases, consistent with the behaviour of at least one Galactic black hole system in the strongly-Comptonised very high state. We find no new evidence supporting the presence of an intermediate-mass black hole in this ULX. ; This is the version of the paper as archived in arXiv.org, subsequently published as Monthly Notices of the Royal Astronomical Society, 2006, 371, pp.1877-1890, available from http://www.blackwell-synergy.com/doi/abs/10.1111/j.1365-2966.2006.10821.x. The definitive version is available from www.blackwell-synergy.com.