  We compute the Morse index of $1$-spike solutions of the semilinear elliptic
problem \begin{equation}\label{abstr} \tag{$\mathcal P_p$} \begin{cases}
-\Delta u= u^p & \text{in $\Omega$} \\ u=0 & \text{on $\partial\Omega$} \\ u>0
& \text{in $\Omega$.} \end{cases} \end{equation}
  where $\Omega\subset \mathbb{R}^2$ is a smooth bounded domain and $p>1$ is
sufficiently large.
  When $\Omega$ is convex, our result, combined with the characterization in
[22], a result in [41] and with recent uniform estimates in \cite{Sirakov},
gives the uniqueness of the solution to \eqref{abstr}, for $p$ large.
  This proves, in dimension two and for $p$ large, a conjecture by
Gidas-Ni-Nirenberg [29].
