  Abridged: The thermal structure of a starless core is crucial for our
understanding of the physics in these objects and hence for our understanding
of star formation. Theory predicts a gas temperature drop in the inner 5000 AU
of these objects, but there has been little observational proof of this. We
performed VLA observations of the NH3 (1,1) and (2,2) transitions towards the
pre-stellar core L 1544 in order to measure the temperature gradient between
the high density core nucleus and the surrounding core envelope. Our VLA
observation for the first time provide measurements of gas temperature in a
core with a resolution smaller than 1000 AU. We have also obtained high
resolution Plateau de Bure observations of the 110 GHz 111-101 para-NH2D line
in order to further constrain the physical parameters of the high density
nucleus. We have estimated the temperature gradient using a model of the source
to fit our data in the u,v plane. We find that indeed the temperature decreases
toward the core nucleus from 12 K down to 5.5 K resulting in an increase of a
factor of 50% in the estimated density of the core from the dust continuum if
compared with the estimates done with constant temperature of 8.75 K. We also
found a remarkably high abundance of deuterated ammonia with respect to the
ammonia abundance (50%+-20%), which proves the persistence of nitrogen bearing
molecules at very high densities (2e6 cm-3) and shows that high-resolution
observations yield higher deuteration values than single-dish observations. Our
analysis of the NH3 and NH2D kinematic fields shows a decrease of specific
angular momentum from the large scales to the small scales.
