  A CAT(0) space has rank at least $n$ if every geodesic lies in an $n$-flat.
Ballmann's Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank
at least $2$ with a geometric group action is rigid -- isometric to a
Riemannian symmetric space, a Euclidean building, or splits as a metric
product. This paper is the first in a series motivated by Ballmann's
conjecture. Here we prove that a CAT(0) space of rank at least $n\geq 2$ is
rigid if it contains a periodic $n$-flat and its Tits boundary has dimension
$(n-1)$. This does not require a geometric group action. The result relies
essentially on the study of flats which do not bound flat half-spaces --
so-called Morse flats. We show that the Tits boundary $\partial_T F$ of a
periodic Morse $n$-flat $F$ contains a regular point -- a point with a
Tits-neighborhood entirely contained in $\partial_T F$. More precisely, we show
that the set of singular points in $\partial_T F$ can be covered by finitely
many round spheres of positive codimension.
