NGC 1407 is the central elliptical in a nearby evolved group of galaxies apparently destined to become a galaxy cluster core. We use the kinematics of globular clusters (GCs) to probe the dynamics and mass profile of the group's center, out to a radius of 60 kpc (~10 galaxy effective radii)—the most extended data set to date around an early-type galaxy. This sample consists of 172 GC line-of-sight velocities, most of them newly obtained using Keck/DEIMOS, with a few additional objects identified as dwarf-globular transition objects or as intragroup GCs. We find weak rotation for the outer parts of the GC system (v/σ ~ 0.2), with a rotational misalignment between the metal-poor and metal-rich GCs. The velocity dispersion profile declines rapidly to a radius of ~20 kpc, and then becomes flat or rising to ~60 kpc. There is evidence that the GC orbits have a tangential bias that is strongest for the metal-poor GCs—in possible contradiction to theoretical expectations. We construct cosmologically motivated galaxy+dark halo dynamical models and infer a total mass within 60 kpc of ~3 × 10¹² M _☉, which extrapolates to a virial mass of ~6 × 10¹³ M _☉ for a typical lambda cold dark matter (ΛCDM) halo—in agreement with results from kinematics of the group galaxies. We present an independent Chandra-based analysis, whose relatively high mass at ~20 kpc disagrees strongly with the GC-based result unless the GCs are assumed to have a peculiar orbit distribution, and we therefore discuss more generally some comparisons between X-ray and optical results. The group's B-band mass-to-light ratio of ~800 (uncertain by a factor of ~2) in Solar units is extreme even for a rich galaxy cluster, much less a poor group—placing it among the most dark matter (DM) dominated systems in the universe, and also suggesting a massive reservoir of baryons lurking in an unseen phase, in addition to the nonbaryonic DM. We compare the kinematical and mass properties of the NGC 1407 group to other nearby groups and clusters, and discuss some implications of this system for structure formation.