  The Arecibo Ultra Deep Survey (AUDS) combines the unique sensitivity of the
telescope with the wide field of the Arecibo L-band Feed Array (ALFA) to
directly detect 21cm HI emission from galaxies at distances beyond the local
Universe bounded by the lower frequency limit of ALFA (z=0.16). AUDS has
collected 700 hours of integration time in two fields with a combined area of
1.35 square degrees. In this paper we present data from 60% of the total
survey, corresponding to a sensitivity level of 80 micro-Jy. We discuss the
data reduction, the search for galaxies, parametrisation, optical
identification and completeness. We detect 102 galaxies in the mass range of
log M_HI/M_sun-2log h=5.6-10.3. We compute the HI mass function (HIMF) at the
highest redshifts so far measured. A fit of a Schechter function results in
alpha=-1.37+-0.03, Phi=(7.72+-1.4)*10^3 h^3/Mpc^3 and log
M_HI/M_sun=9.75+-0.041+2log h. Using the measured HIMF, we find a cosmic HI
density of Omega_HI=(2.33+-0.07)*10^-4/h for the sample z=0.065. We discuss
further uncertainties arising from cosmic variance. Because of its depth, AUDS
is the first survey that can determine parameters for the HI mass function in
independent redshift bins from a single homogeneous data set. The results
indicate little evolution of the co-moving mass function and Omega_HI within
this redshift range. We calculate a weighted average for Omega_HI in the range
$0<z<0.2$, combining the results from AUDS as well as results from other 21cm
surveys and stacking, finding a best combined estimate of
Omega_HI=(2.63+-0.10)*10-4/h.
