  We plan to analyse dust extinction in Lyman Break Galaxies (LBGs) by
introducing a new and more reliable approach to their selection and to the
characterization of their distribution of UV slopes beta, using deep IR images
from HST. We exploit deep WFC3 IR observations of the ERS and HUDF fields over
GOODS-South, combined with HST-ACS optical data, to select z~4 LBGs through a
new (B-V) vs. (V-H) colour diagram. The UV slope of the selected galaxies is
robustly determined by a linear fit over their observed I, Z, Y, J magnitudes,
coherently with the original definition of beta. The same fit is used to
determine their rest-frame UV magnitudes M1600 through a simple interpolation.
We estimate the effect of observational uncertainties with detailed simulations
that we also exploit, under a parametric maximum-likelihood approach, to
constrain the probability density function of UV slopes PDF(beta) as a function
of rest-frame magnitude. We find 142 and 25 robust LBGs in the ERS and HUDF
fields respectively, limiting our sample to S/N(H)>10 objects. Our newly
defined criteria improve the selection of z~4 LBGs and allow us to exclude red
interlopers at lower redshift, especially z~3-3.5 objects. We find that z~4
LBGs are characterized by blue UV slopes, suggesting a low dust extinction: all
L<L* galaxies have an average UV slope <beta>\simeq-2.1, while brighter objects
only are slightly redder (<beta> -1.9). We find an intrinsic dispersion ~ 0.3
for PDF(beta) at all magnitudes. The SFRD at z~4 corrected according to these
estimates turns out to be lower than previously found: log(SFRD)\simeq-1.09
M_sun/yr/Mpc^3. Finally, we discuss how the UV slope of z~4 galaxies changes as
a function of the dust-corrected UV magnitude (i.e. SFR) showing that most
galaxies with a high SFR (> 80 M_sun/yr) are highly extincted objects.
[Abridged]
