  To each partition $\lambda$ with distinct parts we assign the probability
$Q_\lambda(x) P_\lambda(y)/Z$ where $Q_\lambda$ and $P_\lambda$ are the Schur
$Q$-functions and $Z$ is a normalization constant. This measure, which we call
the shifted Schur measure, is analogous to the much-studied Schur measure. For
the specialization of the first $m$ coordinates of $x$ and the first $n$
coordinates of $y$ equal to $\alpha$ ($0<\alpha<1$) and the rest equal to zero,
we derive a limit law for $\lambda_1$ as $m,n\ra\infty$ with $\tau=m/n$ fixed.
For the Schur measure the $\alpha$-specialization limit law was derived by
Johansson. Our main result implies that the two limit laws are identical.
