  We present a complete classification of Einstein metrics on the space M = I
\times S^3, where I is the interval (0,l) or (0,\infty) or their closures, and
we consider separate metric functions f and h (functions of I) for the base and
fiber of the Hopf fibration S^1 -> S^3 -> S^2. All such metrics yielding smooth
and complete manifolds are included and discussed. The results are surprisingly
rich, including many well-known examples and several one-parameter families of
metrics with a variety of geometries.
