A complete continuous time formulation of the Regge calculus is presented by developing the associated continuous time Regge action. It will be shown that the time constraint is, by way of the Bianchi identities (which will also be developed in detail), conserved by the evolution equations. This analysis will also lead to an explicit first integral for each of the evolution equations. The dynamical equations of the theory will therefore be reduced to a set of first-order differential equations. In this formalism the time constraints will reduce to a simple sum of the integration constants. This result is unique to the Regge calculus-there does not appear to be a complete set of first integrals available for the vacuum Einstein equations.