  We prove a positive mass theorem for $n$-dimensional asymptotically flat
manifolds with a non-compact boundary if either $3\leq n\leq 7$ or if $n\geq 3$
and the manifold is spin. This settles, for this class of manifolds, a question
posed in a recent paper by the first author in connection with the long-term
behavior of a certain Yamabe-type flow on scalar-flat compact manifolds with
boundary.
