Gaussian Scale Mixture Representation of $\exp(-\|x\|_2)$

Question

Consider the probability measure $p$ defined on $\mathbf{R}^d$ with density

$$p (x) \propto \exp(-\|x\|_2).$$

I want to know

1. Whether $p$ can be expressed as a Gaussian scale mixture, i.e. whether there exists some probability measure $\pi$ such that

$$p (x) = \int_{\tau > 0} \pi (\tau) \mathcal{N} \left( x | 0, \tau^{-1} I_d \right) d\tau.$$

2. If so, I would like to know the form of $\pi$, if it is known explicitly.

It seems reasonably likely (to me) that (1) should be true and (2) should be known, but I haven't been able to find any appropriate references. Any help would be greatly appreciated.