Rapidity-odd directed flow($v_1$) measurements for charged pions, protons and antiprotons near mid-rapidity ($y=0$) are reported in $\sqrt{s_{NN}} =$ 7.7, 11.5, 19.6, 27, 39, 62.4 and 200 GeV Au + Au collisions as recorded by the STAR detector at the Relativistic Heavy Ion Collider (RHIC). At intermediate impact parameters, the proton and net-proton slope parameter $dv_1/dy|_{y=0}$ shows a minimum between 11.5 and 19.6 GeV. In addition, the net-proton $dv_1/dy|_{y=0}$ changes sign twice between 7.7 and 39 GeV. The proton and net-proton results qualitatively resemble predictions of a hydrodynamic model with a first-order phase transition from hadronic matter to deconfined matter, and differ from hadronic transport calculations.