  We show that a nef and big line bundle whose adjoint bundle has non-zero
global sections on a nonsingular toric weak Fano 3-fold is normally generated.
As a consequence, we see that all ample line bundles on a nonsingular toric
weak Fano 3-fold are normally generated. We also show that an ample line bundle
whose adjoint bundle has non-zero global sections on a Gorensein toric Fano
3-fold is normally generated. In our proof we do not use full classifications
of Fano polytopes.
