  The cusp-core problem is one of the main challenges of the cold dark matter
paradigm on small scales: the density of a dark matter halo is predicted to
rise rapidly toward the center as rho ~ r^alpha with alpha between -1 and -1.5,
while such a cuspy profile has not been clearly observed. We have carried out
the spatially-resolved mapping of gas dynamics toward a nearby ultra-diffuse
galaxy (UDG), AGC 242019. The derived rotation curve of dark matter is well
fitted by the cuspy profile as described by the Navarro-Frenk-White model,
while the cored profiles including both the pseudo-isothermal and Burkert
models are excluded. The halo has alpha=-(0.90+-0.08) at the innermost radius
of 0.67 kpc, Mhalo=(3.5+-1.2)E10 Msun and a small concentration of 2.0+-0.36.
AGC 242019 challenges alternatives of cold dark matter by constraining the
particle mass of fuzzy dark matter to be < 0.11E-22 eV or > 3.3E-22 eV , the
cross section of self-interacting dark matter to be < 1.63 cm2/g, and the
particle mass of warm dark matter to be > 0.23 keV, all of which are in tension
with other constraints. The modified Newtonian dynamics is also inconsistent
with a shallow radial acceleration relationship of AGC 242019. For the feedback
scenario that transforms a cusp to a core, AGC 242019 disagrees with the
stellar-to-halo-mass-ratio dependent model, but agrees with the
star-formation-threshold dependent model. As a UDG, AGC 242019 is in a
dwarf-size halo with weak stellar feedback, late formation time, a normal
baryonic spin and low star formation efficiency (SFR/gas).
