Alexandria's Democratic-controlled City Council voted 6 to 1 last night to appoint J. Harvey Harrison, a moderate Republican, to a one-year term on the city School Board, rejecting for the second time this year a bid by A. Melvin Miller, a government administrator.

Lionel R. Hope, the only black member of the council, cast the lone vote for Miller and said afterward that he was "disappointed and frustrated" at the council's action. "We need another qualified black on the School Board and Melvin was it. I think qualifications should have overruled politics on this one."

Two of the board's nine members are blacks and community leaders have sought to increase the number of minorities on the board, citing the school system's enrollment, which is almost half black.

The fact that Miller, an administrator in the Department of Housing and Urban Development, has worked in the campaigns of local Republicans, and once ran as an independent mayoral candidate against a Democrat did not helped his cause last night, according to local politicians and city officials.

Harrison, 55, a Washington computer firm executive, will fill the seat left vacant by board member James Skidmore, who has left the city. Last June, Harrison, then an school board incumbent, and Miller competed unsuccessfully with three other candidates for three board slots. Harrison's failure to garner enough council votes to retain his seat surprised many city officials and some board members.

School Board Chairman Lou B. Cook said last night that she was "very pleased" with the council's vote, and described Harrison's loss this summer as a "misunderstanding. Harvey was a good man."