We continue with the study of the Hankel determinant, defined by D-n(t, alpha) = det. (integral(infinity)(0) x(j+k)w(x; t, alpha)dx)(j, k= 0)(n-1), generated by a singularly perturbed Laguerre weight, w(x; t, alpha) = x(alpha)e(-x)e(-t/x), x is an element of R+, alpha > 0, t > 0, and obtained through a deformation of the Laguerre weight function, w(x; 0, alpha) = x(alpha)e(-x), x is an element of R+, alpha > 0, via themultiplicative factor e(-t/x). An earlier investigation was made on the finite n aspect of such determinants, which appeared in Chen and Its [J. Approx. Theory 162, 270-297 (2010)]. It was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular Painleve III (P-III, for short) transcendent and its t derivatives. In this paper, we show that under a double scaling, where n, the size of the Hankel matrix tends to infinity, and t tends to 0(+), the scaled-and therefore, in some sense, infinite dimensional-Hankel determinant has an integral representation in terms of a C potential. The second order non-linear ordinary differential equation satisfied by C, after a change of variables, is another P-III transcendent, albeit with fewer number of parameters. Expansions of thedouble scaled determinant for small and large parameters are obtained. (C) 2015 AIP Publishing LLC.