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 separa
ted from the centre of NGC 3079 by 64 kpc, in projection. No absorption is
detected to a limit of 3 Delta T-b/(-T-c) = 0.01. We have considered models
for HI + H-2 clouds characterised by the cloud radius and fractal dimensio
n. Using the upper limit on absorption, we have ruled out a limited but int
eresting region of this parameter space. The observations do not rule out t
he 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 fact
ors, then Hi cannot constitute more than similar to 10(-5), by mass, of the
galaxy's dark matter.