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°6° 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%90% C.L. upper limit on the diffuse astrophysical neutrino flux with a value per neutrino flavour of E2·F90%=4.9·10-8GeV·cm-2·s-1·sr-1E2·F90%=4.9·10-8GeV·cm-2·s-1·sr-1 is set, applicable to the energy range from 23 TeV to 7.8 PeV, assuming an unbroken E-2E-2 spectrum and neutrino flavour equipartition at Earth.