We discuss a gauged U(1)(R) supergravity on five-dimensional (5D) orbifold (S-1/Z(2)) in which both a Z(2)-even U(1) gauge field and the Z(2)-odd graviphoton take part in the U(1)(R) gauging. Based on the off-shell formulation of 5D supergravity, we analyze the structure of Fayet-Iliopoulos (FI) terms allowed in such model. Introducing a Z(2)-even U(1)(R) gauge field accompanies new bulk and boundary FI terms in addition to the known integrable boundary FI term which could be present in the absence of any gauged U(1)(R) symmetry. Some physical consequences of these new FI terms are examined.