Increasing evidence suggests that, similar to their low-mass counterparts, high-mass stars form through a disk-mediated accretion process. At the same time, formation of high-mass stars still necessitates high accretion rates, and hence, high gas densities, which in turn can cause disks to become unstable against gravitational fragmentation. We study the kinematics and fragmentation of the disk around the high-mass star forming region AFGL 2591-VLA 3 which was hypothesized to be fragmenting based on the observations that show multiple outflow directions. We use a new set of high-resolution (0.19 arcsec) IRAM/NOEMA observations at 843 micron towards VLA 3 which allow us to resolve its disk, characterize the fragmentation, and study its kinematics. In addition to the 843 micron continuum emission, our spectral setup targets warm dense gas and outflow tracers such as HCN, HC$_3$N and SO$_2$, as well as vibrationally excited HCN lines. The high resolution continuum and line emission maps reveal multiple fragments with subsolar masses within the inner 1000 AU of VLA 3. Furthermore, the velocity field of the inner disk observed at 843 micron shows a similar behavior to that of the larger scale velocity field studied in the CORE project at 1.37 mm. We present the first observational evidence for disk fragmentation towards AFGL 2591-VLA 3, a source that was thought to be a single high-mass core. While the fragments themselves are low-mass, the rotation of the disk is dominated by the protostar with a mass of 10.3$\pm 1.8~M_{\odot}$. These data also show that NOEMA Band 4 can obtain the highest currently achievable spatial resolution at (sub-)mm wavelengths in observations of strong northern sources.Read less <