A search for heavy neutral leptons has been performed with the ArgoNeuT detector exposed to the NuMI neutrino beam at Fermilab. We search for the decay signature N→νμ^{+}μ^{-}, considering decays occurring both inside ArgoNeuT and in the upstream cavern. In the data, corresponding to an exposure to 1.25×10^{20} POT, zero passing events are observed consistent with the expected background. This measurement leads to a new constraint at 90% confidence level on the mixing angle |U_{τN}|^{2} of tau-coupled Dirac heavy neutral leptons with masses m_{N}=280-970 MeV, assuming |U_{eN}|^{2}=|U_{μN}|^{2}=0.