To each partition $\lambda=(\lambda_1,\lambda_2,\ldots)$ 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 $m,n \to \infty$ with
$\tau=m/n$ fixed. For the Schur measure, the
$\alpha$-specialization limit law was derived by Johansson
[J1]. Our main result implies that the two limit laws are
identical.