  We present an algorithm to compute the exact probability $R_{n}(p)$ for a
site percolation cluster to span an $n\times n$ square lattice at occupancy
$p$. The algorithm has time and space complexity $O(\lambda^n)$ with $\lambda
\approx 2.6$. It allows us to compute $R_{n}(p)$ up to $n=24$. We use the data
to compute estimates for the percolation threshold $p_c$ that are several
orders of magnitude more precise than estimates based on Monte-Carlo
simulations.
