Empirical evidence has emerged of the possibility of fractional cointegration such that thegap, ß, between the integration order d of observable time series, and the integrationorder ? of cointegrating errors, is less than 0.5. This includes circumstances whenobservables are stationary or asymptotically stationary with long memory (so d < 1/2),and when they are nonstationary (so d 1/2). This "weak cointegration" contrastsstrongly with the traditional econometric prescription of unit root observables and shortmemory cointegrating errors, where ß = 1. Asymptotic inferential theory also differs fromthis case, and from other members of the class ß > 1/2, in particular=consistent - n andasymptotically normal estimation of the cointegrating vector ? is possible when ß < 1/2,as we explore in a simple bivariate model. The estimate depends on ? and d or, morerealistically, on estimates of unknown ? and d. These latter estimates need to beconsistent - n , and the asymptotic distribution of the estimate of ? is sensitive to theirprecise form. We propose estimates of ? and d that are computationally relativelyconvenient, relying on only univariate nonlinear optimization. Finite sample performanceof the methods is examined by means of Monte Carlo simulations, and severalapplications to empirical data included.