Successful management of a software project, besides a well-defined project development process, requires an early estimate of project complexity. In a prevailing practice, software development costs usually have been determined a posteriori i.e. after software project implementation. It is essential, however, to know this estimate a priori, i.e., before commencement of works. This paper presents an attempt to construct a methodology that would enable an early estimate of software development cost and its refinements during subsequent development phases. The methodology assumes an object-oriented approach based on the Unified Modeling Language (UML) and Unified Software Development Process (USDP). It outlines an Use Case Driven, Architecture-Centric, Iterative and Incremental estimate process that could significantly improve and simplify early cost estimates. The presented methodology is illustrated on example of the POST software development project.