  A novel algorithm to reconstruct neutrino-induced particle showers within the
ANTARES neutrino telescope is presented. The method achieves a median angular
resolution of $6^\circ$ for shower energies below 100 TeV. Applying this
algorithm to 6 years of data taken with the ANTARES detector, 8 events with
reconstructed shower energies above 10 TeV are observed. This is consistent
with the expectation of about 5 events from atmospheric backgrounds, but also
compatible with diffuse astrophysical flux measurements by the IceCube
collaboration, from which 2 - 4 additional events are expected. A 90% C.L.
upper limit on the diffuse astrophysical neutrino flux with a value per
neutrino flavour of $E^2\cdot \Phi^{90\%} = 4.9 \cdot 10^{-8}$ GeV $\cdot$
cm$^{-2} \cdot$ s$^{-1} \cdot$ sr$^{-1}$ is set, applicable to the energy range
from 23 TeV to 7.8 PeV, assuming an unbroken $E^{-2}$ spectrum and neutrino
flavour equipartition at Earth.
