  A planar pure braid consists of $n$ descending smooth arcs, each connecting a
point on one horizontal line $\ell_{1}$ to a point on a horizontal line
$\ell_{2}$, which is required to be directly below the first point. Two arcs
are allowed to cross, but no threefold intersections are allowed. The set
$\Gamma_{n}$ of all planar pure braids on $n$ strands is a group with respect
to a natural stacking operation.
  We show that $\Gamma_{n}$ is always a diagram group, in the sense of Guba and
Sapir. A number of consequences follow, including biautomaticity and
bi-orderability of the groups $\Gamma_{n}$. Moreover, each group $\Gamma_{n}$
acts properly and cocompactly on a CAT(0) cubical complex.
  (The current version corrects a typographical error and acknowledges overlap
with earlier work of Genevois.)
