  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 $N \to \nu \mu^+ \mu^-$, considering decays occurring both inside
ArgoNeuT and in the upstream cavern. In the data, corresponding to an exposure
to $1.25 \times 10^{20}$ 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 $\left\vert U_{\tau N}\right\rvert^2$ of
tau-coupled Dirac Heavy Neutral Leptons with masses $m_N =$ 280 - 970 MeV,
assuming $\left\vert U_{eN}\right\rvert^2 = \left\vert U_{\mu N}\right\rvert^2
= 0$.
