  We present an analysis of the rotational velocity of the primary of LB 3459
based on 107 new high-resolution and high-S/N ESO VLT UVES spectra. 105 of them
cover a complete orbital period (0.26 d) of this binary system. We have
determined an orbital period of P = 22600.702 +/- 0.005 sec, a radial velocity
amplitude of A_1 = 39.19 +/- 0.05 km/sec, and T_0 = 2451917.152690 +/-
0.000005. From simulations of the He II 4686A line profile (based on NLTE model
atmosphere calculations), we derive v_rot = 47 +/- 5 km/sec.
  We present an animation which shows the orbital movement of the binary
system, its synthetic lightcurve, and compares the phase-dependent variation of
the predicted with the observed He II 4686A line profile.
  The radius of the cool component is almost the same size like Jupiter but its
mass 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 the
brown-dwarf mass range (0.013 - 0.08 M_sun). It might be a former planet which
has survived the previous common-envelope phase and even has gained mass.
