We construct the Weyl multiplets of N = 2 conformal supergravity in five di
mensions. We show that there exist two different versions of the Weyl multi
plet, which contain the same gauge fields but differ in the matter field co
ntent: the Standard Weyl multiplet and the Dilaton Weyl multiplet. At the l
inearized level we obtain the transformation rules for the Dilaton Weyl mul
tiplet by coupling it to the multiplet of currents corresponding to an on-s
hell vector multiplet. We construct the full non-linear transformation rule
s for both multiplets by gauging the D = 5 superconformal algebra F-2(4). W
e show that the Dilaton Weyl multiplet can also be obtained by solving the
equations of motion for an improved vector multiplet coupled to the Standar
d Weyl multiplet.