A Lagrange multiplier test of the null hypothesis of cointegration in fractionally cointegrated models is proposed. The test statistic uses fully modified residuals to cancel the endogeneity and serial correlation biases, and standard asymptotic properties apply under the null and under local alternatives. With iid Gaussian errors, the asymptotic Gaussian power envelope of all (unbiased) tests is achieved by the one-sided (two-sided) test. The finite-sample properties are illustrated by a Monte Carlo study. In an application to the dynamics among exchange rates for seven major currencies against the U.S. dollar, mixed evidence of the existence of a cointegrating relation is found.