State Offices of Rural Health Amendments of 1993 - Revises matching funds provisions of the program of grants to States for operation of offices of rural health (under the Public Health Service Act) to prohibit the Secretary of Health and Human Services: (1) from making grants under the program unless the State agrees to provide non-Federal contributions towards specified costs, in cash, in an amount not less than one dollar for each dollar of Federal funds provided in the grant; and (2) in determining the amount of non-Federal contributions in cash that a State has provided, from including any amounts provided to the State by the Federal Government. Authorizes appropriations. Increases grant amounts that may be made before termination of the program.