  A base B for a finite permutation group G acting on a set X is a subset of X
with the property that only the identity of G can fix every point of B. We
prove that a primitive diagonal group G has a base of size 2 unless the top
group of G is the alternating or symmetric group acting naturally, in which
case a tight bound for the minimal base size of G is given. This bound also
satisfies a well-known conjecture of Pyber. Moreover, we prove that if the top
group of G does not contain the alternating group, then the proportion of pairs
of points that are bases for G tends to 1 as |G| tends to infinity. A similar
result for the case when the degree of the top group is fixed is given.
