Breadth-first heuristic search (BFHS) is a classic algorithm for optimally solving heuristic search and planning problems. BFHS is slower than A* but requires less memory. However, BFHS only works on unit-cost domains. We propose a new algorithm that extends BFHS to domains with different edge costs, which we call uniform-cost heuristic search (UCHS). Experimental results show that the iterative-deepening version of UCHS, IDUCHS, is slower than A* but requires less memory on a variety of planning domains.