  Pulsating thermal X-ray emission from millisecond pulsars can be used to
obtain constraints on the neutron star equation of state, but to date only five
such sources have been identified. Of these five millisecond pulsars, only two
have well constrained neutron star masses, which improve the determination of
the radius via modelling of the X-ray waveform. We aim to find other
millisecond pulsars that already have well constrained mass and distance
measurements that show pulsed thermal X-ray emission in order to obtain tight
constraints on the neutron star equation of state. The millisecond pulsar
PSR~J1909--3744 has an accurately determined mass, M = 1.54$\pm$0.03 M$_\odot$
(1 $\sigma$ error) and distance, D = 1.07$\pm$0.04 kpc. We analysed {\em
XMM-Newton} data of this 2.95 ms pulsar to identify the nature of the X-ray
emission. We show that the X-ray emission from PSR~J1909--3744 appears to be
dominated by thermal emission from the polar cap. Only a single component model
is required to fit the data. The black-body temperature of this emission is
kT=0.26\ud{0.03}{0.02} keV and we find a 0.2--10 keV un-absorbed flux of 1.1
$\times$ 10$^{-14}$ erg cm$^{-2}$ s$^{-1}$ or an un-absorbed luminosity of 1.5
$\times$ 10$^{30}$ erg s$^{-1}$. Thanks to the previously determined mass and
distance constraints of the neutron star PSR~J1909--3744, and its predominantly
thermal emission, deep observations of this object with future X-ray facilities
should provide useful constraints on the neutron star equation of state.
