  [Abridged] The nature and evolutionary properties of the faint radio
population, responsible for the steepening observed in the 1.4 GHz source
counts below 1 milliJy, are not yet entirely clear. Radio spectral indices may
help to constrain the origin of the radio emission in such faint radio sources
and may be fundamental in understanding eventual links to the optical light. We
study the spectral index behaviour of sources that were found in the 1.4 GHz
ATESP survey (Prandoni et al. 2000a,b). Using the Australia Telescope Compact
Array we observed at 5 GHz part of the region covered by the sub-mJy ATESP
survey. In particular we imaged a one square degree area for which deep optical
imaging in UBVRIJK is available. In this paper we present the 5 GHz survey and
source catalogue, we derive the 5 GHz source counts and we discuss the 1.4-5
GHz spectral index properties of the ATESP sources. The analysis of the optical
properties of the sample will be the subject of a following paper. The 5 GHz
survey has produced a catalogue of 111 radio sources, complete down to a (6
sigma) limit S_lim(5 GHz)~0.4 mJy. We take advantage of the better spatial
resolution at 5 GHz (~2 arcsec compared to ~8 arcsec at 1.4 GHz) to infer radio
source structures and sizes. The 5 GHz source counts derived by the present
sample are consistent with those reported in the literature, but improve
significantly the statistics in the flux range 0.4< S(5 GHz)<1 mJy. The ATESP
sources show a flattening of the 1.4-5 GHz spectral index with decreasing flux
density, which is particularly significant for the 5 GHz selected sample. Such
a flattening confirm previous results coming from smaller samples and is
consistent with a flattening of the 5 GHz source counts occurring at fluxes
<=0.5 mJy.
