We report on the first search for nuclear recoils from dark matter in the form of weakly interacting massive particles (WIMPs) with the XENONnT experiment, which is based on a two-phase time projection chamber with a sensitive liquid xenon mass of 5.9 ton. During the $(1.09\pm 0.03)\mathrm{ton}\mathrm{yr}$ exposure used for this search, the intrinsic ${}^{85}\mathrm{Kr}$ and ${}^{222}\mathrm{Rn}$ concentrations in the liquid target are reduced to unprecedentedly low levels, giving an electronic recoil background rate of $(15.8\pm 1.3)\text{events}/\mathrm{ton}\mathrm{yr}\mathrm{keV}$ in the region of interest. A blind analysis of nuclear recoil events with energies between 3.3 and 60.5 keV finds no significant excess. This leads to a minimum upper limit on the spin-independent WIMP-nucleon cross section of $2.58\times {10}^{-47}{\mathrm{cm}}^2$ for a WIMP mass of $28\mathrm{GeV}/c^2$ at 90% confidence level. Limits for spin-dependent interactions are also provided. Both the limit and the sensitivity for the full range of WIMP masses analyzed here improve on previous results obtained with the XENON1T experiment for the same exposure.