Let be a finite extension of , let be a finite abelian Galois extension of odd degree and let be the valuation ring of . We define to be the unique fractional -ideal with square equal to the inverse different of . Combining a result of Erez with a result of Fainsilber and Morales we can see that admits an integral normal basis that is self-dual with respect to the trace form if and only if is at most weakly ramified. For an odd prime and contained in certain cyclotomic extensions, Erez has described such self-dual integral normal bases for . Assuming to be unramified we generate odd abelian weakly ramified extensions of using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions. These constructions generalise Erez's results for cyclotomic extensions.