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We present the UDF05 project, a HST Large Program of deep ACS (F606W, F775W,F850LP) and NICMOS (F110W, F160W) imaging of three fields, two of whichcoincide with the NICP1-4 NICMOS parallel observations of the Hubble Ultra DeepField (HUDF). In this first paper we use the ACS data for the NICP12 field, aswell as the original HUDF ACS data, to measure the UV Luminosity Function (LF)of z~5 Lyman Break Galaxies (LBGs) down to very faint levels. Specifically,based on a V-i, i-z selection criterion, we identify a sample of 101 and 133candidate z~5 galaxies down to z_{850}=28.5 and 29.25 magnitudes in the NICP12and in the HUDF fields, respectively. Using an extensive set of Monte Carlosimulations we derive corrections for observational biases and selectioneffects, and construct the rest-frame 1400 A LBG LF over the rangeM_{1400}=[-21.4, -17.1], i.e. down to ~0.04 L* at z~5, and complement it withdata from the Subaru Deep Field (SDF) from Yoshida et al. (2006) to extend itto the brighter end (M_{1400}>-22.2). We show that: (i) Different assumptionsregarding the SED distribution of the LBG population, dust properties andintergalactic absorption result in a 25% variation in the number density ofLBGs at z~5; (ii) Under consistent assumptions for dust properties andintergalactic absorption, the HUDF is about 30% under-dense in z~5 LBGsrelative to the NICP12 field, a variation which is well explained by cosmicvariance; (iii) The faint-end slope of the LF is independent of the specificassumptions for the input physical parameters, and has a value of alpha ~ -1.6,similar to the faint-end slope of the LF that has been measured for LBGs at z~3and z~6.

The UDF05 Follow-up of the HUDF: I. The Faint-End Slope of the Lyman-Break Galaxy Population at z~5