  We study codimension one (transversally oriented) foliations $\fa$ on
oriented closed manifolds $M$ having non-empty compact singular set
$\sing(\fa)$ which is locally defined by Bott-Morse functions. We prove that if
the transverse type of $\fa$ at each singular point is a center and $\fa$ has a
compact leaf with finite fundamental group or a component of $\sing(\fa)$ has
codimension $\ge 3$ and finite fundamental group, then all leaves of $\fa$ are
compact and diffeomorphic, $\sing(\fa)$ consists of two connected components,
and there is a Bott-Morse function $f:M \to [0,1]$ such that $f\colon M
\setminus \sing(\fa) \to (0,1)$ is a fiber bundle defining $\fa$ and
$\sing(\fa) = f^{-1}(\{0,1\})$. This yields to a topological description of the
type of leaves that appear in these foliations, and also the type of manifolds
admiting such foliations. These results unify, and generalize, well known
results for cohomogeneity one isometric actions and a theorem of Reeb for
foliations with Morse singularities of center type. In this case each leaf of
$\fa$ is a sphere fiber bundle over each component of $\sing(\fa)$.
