We present the first dynamical study of the globular cluster system of NGC 4636, the southernmost giant elliptical galaxy of the Virgo cluster, which from X-ray studies has the reputation of being extremely dark matter dominated. Several hundred medium resolution spectra were acquired at the VLT with FORS2/MXU. We obtained velocities for 176 globular clusters. Assuming a distance of 15 Mpc, the clusters are found at projected galactocentric distances in the range 4 to 70 kpc, the overwhelming majority within 30 kpc. Out to this radius, we find a constant projected velocity dispersion for the blue clusters of sigma~200 km/s. Applying a spherical, isotropic model, we calculate a constant circular velocity of 317±25 km/s. The red clusters are found to have a distinctly different behaviour: at a radius of about 3', the velocity dispersion drops by ~50 km/s to about 170 km/s which then remains constant out to a radius of 7'. The cause might be the steepening of the number density profile at ~3' observed for the red clusters. Adopting a high stellar M/L value from the dynamics of the inner stellar body (Kronawitter et al.), we find that a NFW halo with a low dark matter fraction (of about 15% at one effective radius) approximately yields the required total circular velocity. Lower stellar M/L values increase the dark matter fraction, but do not reproduce the circular velocities of the stellar velocity dispersion. Thus, NGC 4636 does not seem to be extremely dark matter dominated.