  Using a recent breakthrough of Smith, we improve the results of Fouvry and
Kl\"uners on the solubility of the negative Pell equation. Let $\mathcal{D}$
denote the set of fundamental discriminants having no prime factors congruent
to $3$ modulo $4$. Stevenhagen conjectured that the density of $D$ in
$\mathcal{D}$ such that the negative Pell equation $x^2-Dy^2=-1$ is solvable
with $x,y\in\mathbb{Z}$ is $58.1\%$, to the nearest tenth of a percent. By
studying the distribution of the $8$-rank of narrow class groups
$\mathrm{CL}^+(D)$ of $\mathbb{Q}(\sqrt{D})$, we prove that the infimum of this
density is at least $53.8\%$.
