We present an extension of the classic A* search procedure to tabular PCFG parsing. The use of A* search can dramatically reduce the time required to find a best parse by conservatively estimating the probabilities of parse completions. We discuss various estimates and give efficient algorithms for computing them. On average-length Penn treebank sentences, our most detailed estimate reduces the total number of edges processed to less than 3% of that required by exhaustive parsing, and a simpler estimate, which requires less than a minute of pre-computation, reduces the work to less than 5%. Un-like best-first and finite-beam methods for achieving this kind of speed-up, an A* method is guaranteed to find the most likely parse, not just an approximation. Our parser, which is simpler to implement than an upward-propagating best-first parser, is correct for a wide range of parser control strategies and maintains worst-case cubic time.