We present the results of a thorough study of the nucleus of the Sgr dwarf spheroidal galaxy and of the bright globular cluster M54 that resides within the same nucleus (Sgr,N). We have obtained accurate radial velocities and metallicity estimates for 1152 candidate Red Giant Branch stars of Sgr and M54 lying within ~ 9 arcmin from the center of the galaxy, from Keck/DEIMOS and VLT/FLAMES spectra of the infrared Calcium II triplet. Using both velocity and metallicity information we selected two samples of 425 and 321 very-likely members of M54 and of Sgr,N, respectively. The two considered systems display significantly different velocity dispersion profiles: M54 has a steeply decreasing profile from r=0, where sigma= 14.2 km/s, to r=3.5 arcmin where it reaches sigma=5.3 km/s, then it appears to rise again to sigma= 10 km/s at r=7 arcmin. In contrast Sgr,N has a uniformly flat profile at sigma=9.6 km/s over the whole 0 < r < 9 arcmin range. Using data from the literature we show that the velocity dispersion of Sgr remains constant at least out to r ~ 100 arcmin and there is no sign of the transition between the outer flat-luminosity-profile core and the inner nucleus in the velocity profile. These results - together with a re-analysis of the surface brightness profile of Sgr,N and a suite of dedicated N-body simulations - provide very strong support for the hypothesis that the nucleus of Sgr formed independently of M54, which probably plunged to its present position, coincident with Sgr,N, because of significant decay of the original orbit due to dynamical friction. Comment: Accepted for publication by the Astronomical Journal. emulateapj.cls, 26 pag., 21 low resolution figures. A full-resolution color version of the paper can be retrieved from http://www.bo.astro.it/SGR/Sgr_nucleus.ps.gz