  In this paper, we consider the Hankel determinants associated with the
singularly perturbed Laguerre weight $w(x)=x^\alpha e^{-x-t/x}$, $x\in (0,
\infty)$, $t>0$ and $\alpha>0$. When the matrix size $n\to\infty$, we obtain an
asymptotic formula for the Hankel determinants, valid uniformly for $t\in (0,
d]$, $d>0$ fixed. A particular Painlev\'{e} III transcendent is involved in the
approximation, as well as in the large-$n$ asymptotics of the leading
coefficients and recurrence coefficients for the corresponding perturbed
Laguerre polynomials. The derivation is based on the asymptotic results in an
earlier paper of the authors, obtained by using the Deift-Zhou nonlinear
steepest descent method.
