  We consider the standard model extended by heavy right handed fermions
transforming as triplets under SU(2)$_L$, which generate neutrino masses
through the Type-III seesaw mechanism. At energies below their respective mass
scales, the heavy fields get sequentially decoupled to give an effective
dimension-5 operator. Above their mass thresholds, these fields also
participate in the renormalization of the wavefunctions, masses and coupling
constants. We compute the renormalization group evolution of the effective
neutrino mass matrix in this model, with particular emphasis on the threshold
effects. The evolution equations are obtained in a basis of neutrino parameters
where all the quantities are well-defined everywhere, including at $\theta_{13}
= 0$. We also point out the important role of the threshold effects and
Majorana phases in the evolution of mixing angles through illustrative
examples.
