  The $n$-dimensional binary hypercube is the graph whose vertices are the
binary $n$-tuples $\{0, 1\}^n$ and where two vertices are connected by an edge
if they differ at exactly one coordinate. We prove that if the edges are
assigned independent mean 1 exponential costs, the minimum length $T_n$ of a
path from $(0, 0, \dots, 0)$ to $(1, 1, \dots, 1)$ converges in probability to
$\ln(1+\sqrt{2}) \approx 0.881$. It has previously been shown by Fill and
Pemantle (1993) that this so-called first-passage time asymptotically almost
surely satisfies $\ln(1+\sqrt{2}) - o(1) \leq T_n \leq 1+o(1)$, and has been
conjectured to converge in probability by Bollob\'as and Kohayakawa (1997). A
key idea of our proof is to consider a lower bound on Richardson's model,
closely related to the branching process used in the article by Fill and
Pemantle to obtain the bound $T_n \geq \ln\left(1+\sqrt{2}\right)-o(1)$. We
derive an explicit lower bound on the probability that a vertex is infected at
a given time. This result is formulated for a general graph and may be
applicable in a more general setting.
