A parameter identification problem is formulated and solved for a multidimensional parameter of a stochastic system where the parameter appears in the drift term of a stochastic differential equation that has the Brownian motion replaced by a fractional Brownian motion with the Hurst parameter in (0.5,1). These latter fractional Brownian motions seem to be useful models for many physical phenomena where Brownian motion is not appropriate. A different stochastic calculus is required for these processes because they are not semimartingales. A family of estimates is given that arises from a formal application of a least squares algorithm. The strong consistency of the family of estimates is verified.