  Context. HD13724 is a nearby solar-type star at 43.48 $\pm$ 0.06 pc hosting a
long-period low-mass brown dwarf detected with the CORALIE echelle spectrograph
as part of the historical CORALIE radial-velocity search for extra-solar
planets. The companion has a minimum mass of $26.77^{+4.4}_{-2.2}
M_{\mathrm{Jup}}$ and an expected semi-major axis of $\sim$ 240 mas making it a
suitable target for further characterisation with high-contrast imaging, in
particular to measure its inclination, mass, and spectrum and thus establish
its substellar nature. Aims. Using high-contrast imaging with the SPHERE
instrument on the Very Large Telescope (VLT), we are able to directly image a
brown dwarf companion to HD13724 and obtain a low-resolution spectrum. Methods.
We combine the radial-velocity measurements of CORALIE and HARPS taken over two
decades and high contrast imaging from SPHERE to obtain a dynamical mass
estimate. From the SPHERE data we obtain a low resolution spectrum of the
companion from Y to J band, as well as photometric measurements from IRDIS in
the J, H and K bands. Results. Using high-contrast imaging with the SPHERE
instrument at the VLT, we report the first images of a brown dwarf companion to
the host star HD13724. It has an angular separation of 175.6 $\pm$ 4.5 mas and
H-band contrast of $10.61\pm0.16$ mag and, using the age estimate of the star
to be $\sim$1 Gyr, gives an isochronal mass estimate of $\sim$44
$M_{\mathrm{Jup}}$. By combining radial-velocity and imaging data we also
obtain a dynamical mass of $50.5^{+3.3}_{-3.5} M_{\mathrm{Jup}}$. Through
fitting an atmospheric model, we estimate a surface gravity of $\log g = 5.5$
and an effective temperature of 1000K. A comparison of its spectrum with
observed T dwarfs estimates a spectral type of T4 or T4.5, with a T4 object
providing the best fit.
