Countable projective spectra of countable inductive limits, called (PLB)-spaces, of weighted Banach spaces of continuous functions are investigated. It is characterized when the derived projective limit functor vanishes in terms of the sequences of the weights defining the spaces. The locally convex properties of the corresponding projective limits are analyzed, too. Mathematics Subject Classification (2000). Primary 46E10; Secondary 46A13, 46M18, 46M40.Keywords. (PLB)-spaces, derived projective limit functor, weighted spaces of continuous functions. Introduction.In this paper we investigate the structure of spaces of continuous functions defined on a locally compact σ-compact space that can be written as a countable intersection of countable unions of weighted Banach spaces of continuous functions. The spaces we are interested in are examples of (PLB)-spaces, i.e. countable projective limits of countable inductive limits of Banach spaces. (PLB)-spaces constitute a class which is strictly larger than the class of (PLS)-spaces. A locally convex space is a (PLS)-space if it is a countable projective limit of (DFS)-spaces (i.e. of countable inductive limits of Banach spaces with compact linking maps). The class of (PLS)-spaces is the smallest class of locally convex spaces which contains the (DFS)-spaces and is stable under the formation of countable products and closed subspaces, and it contains many natural examples from analysis like the space of distributions, the space of real analytic functions and several spaces of ultradifferentiable functions and ultradistributions. In recent years, the class of (PLS)-spaces has played a prominent role in the applications of abstract functional analysis to linear problems in analysis. These problems include the solvability, existence of solution operators and paramenter dependence of linear partial