The galaxy cluster RX J1347.5–1145 is one of the most X-ray luminous and most massive clusters known. This makes it a prime target for studying issues addressing cluster formation and cosmology. Despite the naive expectation that mass estimation for this cluster should be straightforward (high mass and favorable redshift make it an efficient lens; it is bright in X-rays and appears to be fairly relaxed), some studies have reported very discrepant mass estimates. In this paper we present new high-resolution HST ACS and Chandra X-ray data. The high resolution and sensitivity of ACS enabled us to detect and quantify several new multiple images; we use a total of eight for the strong-lensing analysis. Combining it with shape measurements of weak-lensing sources in the central regions of the cluster, we derive a high-resolution, absolutely calibrated mass map. This map provides the best available quantification of the total mass of the central part of the cluster to date. We compare the reconstructed mass with that inferred from the new Chandra X-ray data. Both mass estimates agree extremely well in the observed region (within 400 h⁻¹₇₀ kpc of the cluster center). In addition we study the major baryonic components (gas and stars) and hence derive the dark matter distribution in the center of the cluster. We find that the dark matter and baryons are both centered on the BCG within the uncertainties (alignment is better <10 kpc). We measure the corresponding dark matter profile and find it consistent with both NFW and cored profiles, indicating that a more extended radial analysis is needed to pinpoint the concentration parameter, and hence the inner slope of the dark matter profile.