We construct the Weyl multiplets of N=2 conformal supergravity in five dimensions. We show that there exist two different versions of the Weyl multiplet, which contain the same gauge fields but differ in the matter field content: the Standard Weyl multiplet and the Dilaton Weyl multiplet. At the linearized level we obtain the transformation rules for the Dilaton Weyl multiplet by coupling it to the multiplet of currents corresponding to an on-shell vector multiplet. We construct the full non-linear transformation rules for both multiplets by gauging the D=5 superconformal algebra F^2(4). We show that the Dilaton Weyl multiplet can also be obtained by solving the equations of motion for an improved vector multiplet coupled to the Standard Weyl multiplet.