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We use Taylor series plus an enclosure of the remainder term to validate the existence of a unique solution for initial value problems in ordinary di erential equations and to compute a coarse enclosure of that solution. The signi cance of this result is its application to Lohner's AWA algorithm for validated solutions, not to the theory of ordinary di erential equations. By using high-order Taylor series in Lohner's Algorithm I, we are able to validate the solution over much longer time steps than is done in the current AWA code. For Lohner's enclosure by polynomials, the enclosures are expensive to compute, but it is easy to check for enclosure. For our enclosure by Taylor series, the enclosures are free because they are already being computed, but checking for enclosure requires 2 n polynomial root ndings. Work is continuing on an implementation that will allow direct computational comparisons of the e ectiveness of the two methods.