Estimation of the autoregressive coefficient $\varrho$ in linear models with first-order autoregressive disturbances has been broadly studied in the literature. Based on C.R. Rao's MINQE-theory, Aza\is et al. (1993) gave a new general approach for computing locally optimum estimators of variance-covariance components in models with non-linear structure of the variance-covariance matrix. As a special case, in the linear model with AR(1) errors, we discuss a new algorithm for computing locally optimum quadratic plus constant invariant estimators of the parameters $\varrho$ and $\sigma^2$, respectively. Moreover, simple iteration of this estimation procedure gives a maximum likelihood estimates of both, the first order parameters, and the variance-covariance components.