We present Very Long Baseline Array (VLBA) HI absorption observations of the core region of the powerful radio galaxy Cygnus A. These data show both broad (FWHM = 231 pm 21 km/s) and narrow (FWHM <30 km/s) velocity width absorption components. The broad velocity absorption shows high opacity on the counter jet, low opacity against the core and no absorption on the jet side. We argue that these results are most naturally explained by a circumnuclear HI absorbing disk orientated roughly perpendicular to the jet axis. We estimate that the HI absorbing gas lies at a radius of ~80 pc has a scale height of about 20 pc, density n > 10^{4} cm^{-3} and total column density in the range 10^{23}-10^{24} cm^{-2}. Models in which the HI absorption is primarily from an atomic or a molecular gas phase can both fit our data. Modelling taking into account the effective beam shows that the broad HI absorbing gas component does not cover the radio core in Cygnus A and therefore does not contribute to the gas column that blocks our view of the hidden quasar nucleus. If however Cygnus A were observed from a different direction, disk gas on ~100 pc radius scales would contribute significantly to the nuclear column density, implying that in some radio galaxies gas on these scales may contribute to the obscuration of the central engine. We argue that the circumnuclear torus in Cygnus A contains too little mass to power the AGN over > 10^{7} yr but that material in the outer HI absorbing gas disk can provide a reservoir to fuel the AGN and replenish torus clouds. The second narrow HI absorption component is significantly redshifted (by 186km/s) with respect to the systemic velocity and probably traces infalling gas which will ultimately fuel the source. [abridged]
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