  Ahlswede and Khachatrian's diametric theorem is a weighted version of their
complete intersection theorem, itself an extension of the $t$-intersecting
Erd\H{o}s-Ko-Rado theorem. Their intersection theorem says that the maximum
size of a family of subsets of $[n] = \{1, \dots, n\}$, every pair of which
intersects in at least $t$ elements, is the size of certain trivially
intersecting families proposed by Frankl. We address a cross intersecting
version of their diametric theorem.
  Two families $\mathcal{A}$ and $\mathcal{B}$ of subsets of $[n]$ are {\em
cross $t$-intersecting} if for every $A \in \mathcal{A}$ and $B \in
\mathcal{B}$, $A$ and $B$ intersect in at least $t$ elements. The $p$-weight of
a $k$ element subset $A$ of $[n]$ is $p^{k}(1-p)^{n-k}$, and the weight of a
family $\mathcal{A}$ is the sum of the weights of its sets. The weight of a
pair of families is the product of the weights of the families.
  The maximum $p$-weight of a $t$-intersecting family depends on the value of
$p$. Ahlswede and Khachatrian showed that for $p$ in the range $[\frac{r}{t +
2r - 1}, \frac{r+1}{t + 2r + 1}]$, the maximum $p$-weight of a $t$-intersecting
family is that of the family $\mathcal{F}^t_r$ consisting of all subsets of
$[n]$ containing at least $t+r$ elements of the set $[t+2r]$.
  In a previous paper we showed a cross $t$-intersecting version of this for
large $t$ in the case that $r = 0$. In this paper, we do the same in the case
that $r = 1$. We show that for $p$ in the range $[\frac{1}{t + 1}, \frac{2}{t +
3}]$ the maximum $p$-weight of a cross $t$-intersecting pair of families, for
$t \geq 200$, is achieved when both families are $\mathcal{F}^t_1$. Further, we
show that except at the endpoints of this range, this is, up to isomorphism,
the only pair of $t$-intersecting families achieving this weight.
