  The dielectric properties of strontium titanate (SrTiO$_3$) have previously
been reported from room temperature to low temperatures with conflicting
results. In this work, precision measurement of the permittivity is undertaken
by simultaneously measuring transverse electric and transverse magnetic
resonant modes within a single crystal. It is unequivocally shown that the
permittivity is isotropic at room temperature with a permittivity of order
$316.3\pm2.2$ by measuring multiple modes of different electric field
polarisations. As the crystal is cooled to 5 K and undergoes well known phase
transitions, we show the material becomes uniaxial anisotropic with the ratio
of the parallel to perpendicular permittivity to the cylinder z-axis of the
sample as high as 2.4 below 6 K.
