This paper obtains a higher-order asymptotic expansion of a class of M-estimators of the regression parameter in a one-parameter linear regression model when the errors form a long-memory moving average. Suitably standardized difference between an M-estimator and the least-squares estimator is shown to have a limiting distribution. The nature of the limiting distribution depends on the range of the dependence parameter θ. For example, in the case of the symmetric common error distribution, if 1/3<θ<1, then a suitably standardized difference between the least absolute deviation and the least-squares estimators converges weakly to a normal distribution. If 0<θ<1/3 then the corresponding limiting distribution is not normal.