The singularity probability of a random symmetric matrix is exponentially small

Let $A$ be drawn uniformly at random from the set of all $n\times n$ symmetric matrices with entries in $\{-1,1\}$. We show that $$\mathbb{P}( \det(A) = 0 ) \leq e^{-cn},$$ where $c>0$ is an absolute constant, thereby resolving a well-known conjecture.