A model independent search for a Heavy Neutral Lepton (HNL), capable of mixing with the $\tau$ neutrino, is presented. A total of 424 $fb^{-1}$ of ~\babar~ data is analyzed. No significant signal is seen. Upper limits at the 95 $\%$ confidence level are set on the extended Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix element, $|U_{\tau 4}|^{2}$, which depend on the HNL mass hypothesis and vary from $2.31 \times 10^{-2}$ to $5.04 \times 10^{-6}$, across the mass range $100 < m_{4} < 1300 ~\text{MeV}/c^{2}$. More stringent limits being placed at higher masses.