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We present an analysis of the rotational velocity of the primary of LB 3459based on 107 new high-resolution and high-S/N ESO VLT UVES spectra. 105 of themcover a complete orbital period (0.26 d) of this binary system. We havedetermined an orbital period of P = 22600.702 +/- 0.005 sec, a radial velocityamplitude of A_1 = 39.19 +/- 0.05 km/sec, and T_0 = 2451917.152690 +/-0.5. From simulations of the He II 4686A line profile (based on NLTE modelatmosphere calculations), we derive v_rot = 47 +/- 5 km/sec.  We present an animation which shows the orbital movement of the binarysystem, its synthetic lightcurve, and compares the phase-dependent variation ofthe predicted with the observed He II 4686A line profile.  The radius of the cool component is almost the same size like Jupiter but itsmass is about 70 times higher than Jupiter's mass. Thus, from its present mass(M_2 = 0.066 M_sun), the secondary of LB 3459 lies formally within thebrown-dwarf mass range (0.013 - 0.08 M_sun). It might be a former planet whichhas survived the previous common-envelope phase and even has gained mass.Comment: 7 pages, 11 Postscript figures, to appear in A&

The rotational velocity of the sdOB primary of the eclipsing binary system LB 3459 (AA Dor)