  Quasars at redshifts $z>6$ are an excellent probe of the formation and
evolution of supermassive black holes in the early Universe. The population of
radio-luminous quasars is of particular interest, as such quasars could
potentially be used to study the neutral intergalactic medium during cosmic
reionisation via H$\,$I 21$\,$cm absorption studies. However, the lack of deep
radio observations of $z>6$ quasars leaves the population poorly constrained,
and suitable candidates for an H$\,$I 21$\,$cm absorption study have yet to be
found. In this work, we present Jansky Very Large Array (VLA) 1$-$2 GHz radio
continuum observations of 138 quasars at redshifts $6.0 \leq z<7.6$. We detect
the radio continuum emission of the $z=6.1$ quasar J1034-1425, with a 1.6 GHz
flux density of $170\pm 36\,\mu$Jy. This quasar is radio-quiet with
radio-loudness, $R \equiv f_{5\text{~GHz}}/f_{\nu,\text{4400 A}} = 2.4\pm0.5$.
In addition, we detect 7 other quasars at z>6, which have previously been
characterised in the literature at these frequencies. Using the full sample, we
estimate the radio-loud fraction to be $3.8^{+6.2}_{-2.4}\%$, where the
uncertainties are 95% confidence intervals. This is lower than recent estimates
of the radio-loud fraction in the literature, but is still marginally
consistent with no redshift evolution of the radio-loud fraction. We explore
the undetected quasar population by stacking their continuum images at their
optical positions and obtain a median stacked flux density of $13.8\pm
3.9~\mu$Jy and luminosity of
$\log{L_{5~\mathrm{GHz}}/(\mathrm{W~Hz}^{-1})}=24.2\pm0.1$.
