In this paper, we propose a new method for estimating query cost in client-server based heterogeneous database management system. The cost estimation parameters are adjusted by an Adaptive Cost Estimation (ACE) module which uses query execution feedback yielding more and more accurate cost estimates. The most important features of ACE are its detailed cost model which accounts for all costs incurred, its rapid convergence to the actual parameter values, and its low overhead which permits continuous adaptation during the run time of the system. ACE has been implemented and tested with Oracle 6, Oracle 7, Ingres, and ADMS. Extensive experiments performed on these systems show that the ACE's time estimates are within 20% of the real wall-clock time for more than 92% of the queries. This percentage surpasses 98% for queries over 20 seconds. (Also cross-referenced as UMIACS-TR-96-37)