Resolving singularities of plane analytic branches with one toric morphism

Let (C, 0) be an irreducible germ of complex plane curve. Let Γ ⊂ N be the semigroup associated to it and CΓ ⊂ Cg+1 the corresponding monomial curve, where g is the number of Puiseux exponents of (C, 0). We show, us-ing the specialization of (C, 0) to (CΓ, 0), that the same toric morphisms Z(Σ) → Cg+1 which induce an embedded resolution of singularities of (CΓ, 0) also resolve the singularities of (C, 0) ⊂ (Cg+1, 0), the embedding being defined by elements of the analytic algebra OC,0 whose valuations generate the semigroup Γ.