We report on the first extra-solar planet discovered with the brand new HARPS instrument. The planet is a typical ``hot Jupiter'' with m2 sin i=0.62 MJup and an orbital period of 3.39 days, but from the photometric follow-up of its parent star HD 330075 we can exclude the presence of a transit. The induced radial-velocity variations exceed 100 m s-1 in semi-amplitude and are easily detected by state-of-the-art spectro-velocimeters. Nevertheless, the faint magnitude of the parent star (V=9.36) benefits from the efficient instrument: with HARPS less than 10 observing nights and 3 h of total integration time were needed to discover the planet and characterize its orbit. The orbital parameters determined from the observations made during the first HARPS run in July 2003 have been confirmed by 7 additional observations carried out in February 2004. The bisector analysis and a photometric follow-up give no hint for activity-induced radial-velocity variations, indicating that the velocity curve is best explained by the presence of a low-mass companion to the star. In this paper we present a set of 21 measurements of excellent quality with weighted rms as low as 2.0 m s-1. These measurements lead to a well defined orbit and consequently to the precise orbital parameters determination of the extra-solar planet HD 330075 b. Based on observations made with the HARPS instrument on the ESO 3.6 m telescope at the La Silla Observatory under programme ID 72.C-0488
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