  In recent years an increasing number of devices and experiments are shown to
be limited by mechanical thermal noise. In particular sub-Hertz laser frequency
stabilization and gravitational wave detectors, that are able to measure
fluctuations of 1E-18 m/rtHz or less, are being limited by thermal noise in the
dielectric coatings deposited on mirrors. In this paper we present a new
measurement of thermal noise in low absorption dielectric coatings deposited on
micro-cantilevers and we compare it with the results obtained from the
mechanical loss measurements. The coating thermal noise is measured on the
widest range of frequencies with the highest signal to noise ratio ever
achieved. In addition we present a novel technique to deduce the coating
mechanical losses from the measurement of the mechanical quality factor which
does not rely on the knowledge of the coating and substrate Young moduli. The
dielectric coatings are deposited by ion beam sputtering. The results presented
here give a frequency independent loss angle of (4.70 $\pm$ 0.2)x1E-4 with a
Young's modulus of 118 GPa for annealed tantala from 10 Hz to 20 kHz. For
as-deposited silica, a weak frequency dependence (~ f^{-0.025}) is observed in
this frequency range, with a Young's modulus of 70 GPa and an internal damping
of (6.0 $\pm$ 0.3)x1E-4 at 16 kHz, but this value decreases by one order of
magnitude after annealing and the frequency dependence disappears.
