Law Enforcement Scholarship and Recruitment Act - Directs each State to pay from funds under this Act the Federal share (not more than 60 percent) of the cost of: (1) awarding scholarships to in-service law enforcement personnel for further education; and (2) providing full-time employment in summer or part-time employment for a period up to a year to students enrolled in accredited institutions of higher education who demonstrate an interest in law enforcement careers. Requires each State receiving such funds to designate a lead agency to conduct such programs. Makes the Director of the Bureau of Justice Assistance responsible for administration of such program and for issuing rules. Provides that such a scholarship shall be for one academic year and may be used at any accredited institution of higher education. Conditions eligibility on an individual's having been employed in law enforcement for two years immediately preceding the date on which scholarship assistance is sought. Makes individuals who have been employed as law enforcement officers ineligible to participate in student employment programs under this Act. Requires each scholarship recipient to work in a law enforcement position in the State that awards the scholarship for a period of one month for each credit hour for which funds are received (with a six-month minimum and two-year maximum). Authorizes and allocates appropriations.