MERIDIAN, Miss. — Across the street from a barbershop and upstairs from a post office sits a big empty room where Mississippi once began to face up to itself.

Even on a steamy humdrum Thursday afternoon, this city’s stately federal courtroom looks like the kind of place where momentous things could happen, as they once did. The legal campaign to integrate the University of Mississippi got under way here in May 1961, and it was here that a local posse of Klansmen who murdered three civil rights workers faced justice at the hands of their neighbors, the first time that had happened in Mississippi.

The court has remained in use over the decades since, though with a lower profile. Soon, however, it will be shut down for good, a victim of its quietness and the fiscal urgencies of Washington....