We prove that the quantifier-free fragment of the theory of character strings with regular language membership constraints and linear integer constraints over string lengths is decidable. We do that by describing a sound, complete and terminating tableaux calculus for that fragment which uses as oracles a decision procedure for linear integer arithmetic and a number of computable functions over regular expressions. A distinguishing feature of this calculus is that it provides a completely algebraic method for solving membership constraints which can be easily integrated into multi-theory SMT solvers. Another is that it can be used to generate symbolic solutions for such constraints, that is, solved forms that provide simple and compact representations of entire sets of complete solutions. The calculus is part of a larger one providing the theoretical foundations of a high performance theory solver for string constraints implemented in the SMT solver CVC4.