The coupled Kapchinskij -Vladimirskij (K-V) envelope equations for a charged particle beam transported by a periodic system of quadrupoles with self-consistent space charge force have previously been solved by various approximate methods, with accuracy ranging &om 1% to 10%. A new method of solution is introduced here, which is based on a double expansion of the beam envelope functions in powers of the focal strength and either the beam's emittance or its dimensionless perveance. This method results in accuracy better than 0.1% for typical lattice and beam parameters when carried through one consistent level of approximation higher than employed in previous work. Several useful quantities, such as the values of the mdepressed tune and the beam's perveance in the limit of vanishing emittance, are represented by very rapidly converging power series in the focal strength, with accuracy of .01% or better.