  Let L/K be a finite Galois extension of complete local fields with finite
residue fields and let G=Gal(L/K). Let G_1 and G_2 be the first and second
ramification groups. Thus L/K is tamely ramified when G_1 is trivial and we say
that L/K is weakly ramified when G_2 is trivial. Let O_L be the valuation ring
of L and let P_L be its maximal ideal. We show that if L/K is weakly ramified
and n is congruent to 1 mod |G_1| then P_L^n is free over the group ring
O_K[G], and we construct an explicit generating element. Under the additional
assumption that L/K is wildly ramified, we then show that every free generator
of P_L over O_K[G] is also a free generator of O_L over its associated order in
the group algebra K[G]. Along the way, we prove a `splitting lemma' for local
fields, which may be of independent interest.
