  Using the quantum Cram\'{e}r-Rao bound from quantum estimation theory, we
derive a fundamental quantum limit on the sensitivity of a temperature
measurement of a thermal astronomical source. This limit is expressed in terms
of the source temperature $T_s$, input spectral bandwidth $\Delta \nu$, and
measurement duration $T$, subject to a long measurement time assumption
$T\Delta \nu \gg 1$. It is valid for any measurement procedure that yields an
unbiased estimate of the source temperature. The limit agrees with the
sensitivity of direct detection or photon counting, and also with that of the
ideal radiometer in the regime $kT_s/h \nu_0\gg 1$ for which the Rayleigh-Jeans
approximation is valid, where $\nu_0$ is the center frequency at which the
radiometer operates. While valid across the electromagnetic spectrum, the limit
is especially relevant for radio astronomy in this regime, since it implies
that no ingenious design or technological improvement can beat an ideal
radiometer for temperature measurement. In this connection, our result refutes
the recent claim of a radio astronomy technique with much-improved sensitivity
over the radiometer (Lieu et al. 2015).
