We have obtained a detailed H-alpha velocity map of the superluminous (approximately 6 x 10(11) L.), infrared galaxy NGC 6240 (z = 0.0245) with the Hawaii Imaging Fabry-Perot Interferometer (HIFI). The kinematic data reveal two dynamical systems (disks 1 and 2) that exhibit radically different rotation and are closely spaced in velocity (almost-equal-to 70 km s-1) and position (almost-equal-to 12", almost-equal-to 5.7 kpc). Disk 1 (P.A. = 45-degrees +/- 5-degrees) is roughly aligned with the major axis of the near-infrared continuum and exhibits flat rotation out to approximately 20" in radius, centered on the double nucleus seen at optical, near-infrared, and radio wavelengths. The rotation turns over at r(t)1 almost-equal-to 7".2 with a peak-to-peak velocity amplitude almost-equal-to 280/sin i1 km s-1, where i1 is the disk inclination. A preliminary analysis suggests i1 congruent-to 70-degrees +/- 7-degrees although the effective beam of the filtered HIFI data (4" FWHM) makes a reliable determination difficult. The structure of disk 2 (P.A. = 155-degrees +/- 3-degrees) is much more remarkable. The rotation curve comprises an unresolved, or marginally resolved, central velocity gradient with a peak-to-peak amplitude of almost-equal-to 800/sin i2 km s-1 within r(t)2 < 2".5, and a faster than Keplerian drop-off outside r(t)2. The peak rotation implies a compact mass M2 greater than 4.5 x 10(10) M./sin2 i2 within a radius < 1.2 kpc. Presently, there are no observations of a luminous, compact counterpart to the core of disk 2 at radio, millimeter, infrared, or X-ray wavelengths, although our data show evidence for a weak optical continuum. We derive strict lower limits on the mass-to-light ratio from UBVRIzK photometry, in particular, M2/L(K) > 3 (M./L. sin2 i2) and M2/L(R) > 60 (M./L. sin2 i2). We believe that the observed gas motions provide strong evidence for the existence of an ultramassive, "dark matter" core or a compact dark object (e.g., dead or underfed quasar) in the NGC 6240 merger system.