Black phosphorus (BP) has recently emerged as a promising two-dimensional direct bandgap semiconducting material. Here, we report the fabrication and the electrical transport measurements of the black phosphorus based field-effect transistor with the Au/Co/MgO as drain and source tunneling contacts. By modulating the back-gate voltage, the multilayer black phosphorus channel exhibits ambipolar characteristics (both n-type and p-type) and the conduction behavior can be switched from hole dominated to electron dominated transport region. In the hole dominated region, we have measured a minimum of Schottky barrier height of 37 meV for Au/Co/MgO contact on BP. Moreover, the transistor ON/OFF (Ion/Ioff) ratio is obtained as large as 107 at 20 K and 105 at 300 K. A systematic study of the temperature and the back-gate voltage dependent conduction properties has been performed to understand the modulation of band structure and the ambipolar behavior. The demonstration of high ON/OFF ratio and low Schottky barrier height by using Au/Co/MgO tunneling contacts reveals a promising potential for spintronics applications with multilayer black phosphorus field-effect transistor.