  Given positive integers k and r where 4 divides k and k/2 \leq r \leq k-1, we
give a minimum r-degree condition that ensures a perfect matching in a
k-uniform hypergraph. This condition is best possible and improves on work of
Pikhurko who gave an asymptotically exact result. Our approach makes use of the
absorbing method, as well as the hypergraph removal lemma and a structural
result of Keevash and Sudakov relating to the Turan number of the expanded
triangle.
