  Let $X$ be a rational elliptic surface with elliptic fibration
$\pi:X\to\Bbb{P}^1$ over an algebraically closed field $k$ of any
characteristic. Given a conic bundle $\varphi:X\to\Bbb{P}^1$ we use numerical
arguments to classify all possible fibers of $\varphi$ and study the interplay
between singular fibers of $\pi$ and $\varphi$.
