The nonparametric maximum likelihood estimate (NPMLE) of a distribution functionG, based on a random sample fromG, under the assumption thatGhas a monotone decreasing densityg, is the least concave majorant of the empirical distribution function (Barlow, Bartholomew, Bremner, and Brunk 1972). This may lead one to believe that the least concave majorant of the Kaplan-Meier (1958) estimate is the NPMLE ofGin the “arbitrary right censorship model”—that is, when the data include right censored observations and no assumption on the censoring mechanism is made. This, however, is not correct, as was pointed out by McNichols and Padgett (1982), and the problem of fully characterizing the solution to this problem has been left open, In this article we provide such a characterization in two steps. First, we show that when the largest observation is censored the NPMLE ofGis not a proper distribution (the nonparametric likelihood function has a supremum but not a maximum). This suggests that as a numeric tool one should restrict the maximization problem to distributions with finite support—say, [0,M] for some very largeM. Second, we describe an iterative scheme with an extremely simple iteration step, which is guaranteed to converge monotonically (in the sense of increased likelihood in each step) to the NPMLE ofGon [0,M]. Unlike the Kaplan-Meier estimate, for which the associated density estimate places mass at a discrete set of points, the method in this article produces directly a proper estimate of the densityg. Of course, the method can also be used with uncensored data, in which case it produces the least concave majorant of the empirical distribution function.