  DNA synthesis in \textit{Xenopus} frog embryos initiates stochastically in
time at many sites (origins) along the chromosome. Stochastic initiation
implies fluctuations in the time to complete and may lead to cell death if
replication takes longer than the cell cycle time ($\approx 25$ min).
Surprisingly, although the typical replication time is about 20 min, \textit{in
vivo} experiments show that replication fails to complete only about 1 in 300
times. How is replication timing accurately controlled despite the
stochasticity? Biologists have proposed two solutions to this
"random-completion problem." The first solution uses randomly located origins
but increases their rate of initiation as S phase proceeds, while the second
uses regularly spaced origins. In this paper, we investigate the
random-completion problem using a type of model first developed to describe the
kinetics of first-order phase transitions. Using methods from the field of
extreme-value statistics, we derive the distribution of replication-completion
times for a finite genome. We then argue that the biologists' first solution to
the problem is not only consistent with experiment but also nearly optimizes
the use of replicative proteins. We also show that spatial regularity in origin
placement does not alter significantly the distribution of replication times
and, thus, is not needed for the control of replication timing.
