We perform the first observational test of dark matter in the form of cold (3 K) fractal clouds, as described by Pfenniger et al. (1994) and Pfenniger & Combes (1994). This is accomplished by probing for HI absorption in the halo of NGC 3079 against the background quasar, Q 0957+561, which is separated from the centre of NGC 3079 by 64 kpc, in projection. No absorption is detected to a limit of 3ΔT
c) = 0·01. We have considered models for HI + H2 clouds characterised by the cloud radius and fractal dimension. Using the upper limit on absorption, we have ruled out a limited but interesting region of this parameter space. The observations do not rule out the possibility that all the dark matter could be hidden in the form of cold fractal clouds. By contrast, if the gas is diffuse with unity filling factors, then HI cannot constitute more than ∼ 10−5, by mass, of the galaxy's dark matter.