The group of thermally emitting isolated neutron stars (INSs) known as the Magnificent Seven (M7) is unique among the various neutron star populations. Crustal heating by means of magnetic field decay and an evolutionary link with magnetars may explain why these objects rotate more slowly and have higher thermal luminosities and magnetic field intensities than standard pulsars of similar age. The third brightest INS, RX J1605.3+3249, is the only object amidst the seven still lacking a detected periodicity. We observed the source with the XMM-Newton Observatory for 60 ks aiming at unveiling the neutron star rotation rate and investigating its spectrum in detail. A periodic signal at P=3.387864(16) s, most likely the neutron star spin period, is detected at the 4-sigma confidence level. The coherent combination of the new data with a past XMM-Newton EPIC-pn observation of the source constrains the pulsar spin-down rate at the 2-sigma confidence level, implying a dipolar magnetic field of B~7.4e13 G. If confirmed, RX J1605.3+3249 would be the neutron star with the highest dipolar field amongst the M7. The spectrum of the source shows evidence of a cool blackbody component, as well as for the presence of two broad absorption features. Furthermore, high-resolution spectroscopy with the RGS cameras confirms the presence of a narrow absorption feature at energy 0.57 keV in the co-added spectrum of the source, also seen in other thermally emitting isolated neutron stars. Phase-resolved spectroscopy, as well as a dedicated observing campaign aimed at determining a timing solution, will give invaluable constraints on the neutron star geometry and will allow one to confirm the high value of spin down, which would place the source closer to a magnetar than any other M7 INS.
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