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 $E^2cdot Phi^{90%} = 4.9 cdot 10^{-8}$ 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.
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