From 15 February 2004, 20:00 UT to 18 February 2004, 01:00 UT, the solar wind density dropped to extremely low values (about 0.35 cm −3). On 17 February, between 17:45 UT and 18:10 UT, the CLUSTER spacecraft cross the dayside magnetopause several times at a large radial distance of about 16 RE. During each of these crossings, the spacecraft detect high speed plasma jets in the dayside magnetopause and boundary layer. These observations are made during a period of southward and dawnward Interplanetary Magnetic Field (IMF). The magnetic shear across the local magnetopause is ∼90° and the magnetosheath beta is very low (∼0.15). We evidence the presence of a magnetic field of a few nT along the magnetopause normal. We also show that the plasma jets, accelerated up to 600 km/s, satisfy the tangential stress balance. These findings strongly suggest that the accelerated jets are due to magnetic reconnection between interplanetary and terrestrial magnetic field lines northward of the satellites. This is confirmed by the analysis of the ion distribution function that exhibits the presence of D shaped distributions and of a reflected ion population as predicted by theory. A quantitative analysis of the reflected ion population reveals that the reconnection process lasts about 30 min in a reconnection site located at a very large distance of several tens RE from the Cluster spacecraft. We also estimate the magnetopause motion and thickness during this event. This paper gives the first experimental study of magnetic reconnection during such rare periods of very low solar wind density. The results are discussed in the frame of magnetospheric response to extremely low solar wind density conditions.