A generalized version of the regular interval theorem of Morse theory is proven using techniques from the theory of Riemannian submersions and conformal deformations. This approach provides an interesting link between Riemamnnian submersions (for real valued functions) and Morse theory.Let f: (M,g) -+ R be a smooth real valued function on a noIl-compact complete connected Riemannian manifold (M,g) such that df is bounded in norm away from zero. By pointwise conformally deforming g to pg, p = I(df l 2 , we show that (M, pg) is a complete Riemannian manifold, arid that f: (M, pg) -R is a surjective Riemannian submersion and a globally trivial fiber bundle over R. In particular, all of the level hypersurfaces of f are diffeomorphic, and M is globally diffeomorphic to the product bundle R x f-1(0) by a diffeomorphism F°: R x f-1 (0) -M that "straightens out" the level hypersurfaces of f.Moreover, we show that (FO)*(pg) is a parameterized Riemannian product manifold on Rx f-' (0), i.e., a product manifold with a metric that varies on the fibers {t} x f-'(0). Also, F°: (R x f-1(0), (F°)*(pg)) --(M, g) is a conformal diffeomnorphism between the Rieiannian manifolds (R x f-l(0), (F°)*(pg)) and (M, g), so that (M, g) is confornially equivalent to a parameterized Riemannian product manifold. The conformIlal diffeomorphisIm F°is an isometry between the Riemannian product manifold (R x f-1 (0), 1 + go) (where go is the metric induced by g on f-1(0)) and (M, g) if and only if ldfll = 1 and Hess f = 0.