Weyl fermions are two-component spinors described by irreducible representations of the Lorentz group. They are massless with well-defined helicity and move with the speed of light. Although such kind of elementary particles have not been found yet in real space, the charge excitations in a certain class of materials, the Weyl semimetals, are predicted to behave exactly the same as the Weyl fermions, leading to exotic properties like chiral anomaly and quantum anomalous Hall effect. Carrying out angle-resolved photoemission spectroscopy on TaP, a Weyl semimetal candidate, in a wide photon energy range, we observed the pairs of Weyl cones in the bulk electronic states and the unclosed Fermi arcs on the Ta-terminated surface, which fully agree with theoretical calculations. Our results unambiguously establish that TaP is a Weyl semimetal with one type of well-separated Weyl nodes locates at the chemical potential, which guarantee that the exotic low-energy excitations near the Weyl nodes dominate the transport properties, and may lead to novel applications in this nontoxic transition metal monophosphide.