A new iterative solution to the statistical adjustment of constrained data sets is derived in this paper. The method is general and may be applied to any weighted least squares problem containing nonlinear equality constraints. Other methods are available to solve this class of problem, but are complicated when unmeasured variables and model parameters are not all observable and the model constraints are not all independent. Of notable exception, however, are the methods of Crowe (1986) and Pai and Fisher (1988), although these implementations require the determination of a matrix projection at each iteration which may be computationally expensive. An alternative solution which makes the pragmatic assumption that the unmeasured variables and model parameters are known with a finite but equal uncertainty is proposed. We then re-formulate the well known data reconciliation solution in the absence of these unknowns to arrive at our new solution; hence the regularization approach. Another procedure for the classification of observable and redundant variables which does not require the explicit computation of the matrix projection is also given. The new algorithm is demonstrated using three illustrative examples previously used in other studies.