No single case banned organized prayer and Bible-reading in public schools
; this was accomplished through a number of Supreme Court cases, beginning with Engel v. Vitale, (1962), that tested the limits of the First Amendment Establishment and Free Exercise Clauses.
In
Engel v. Vitale, 370 U.S. 421 (1962), the Supreme Court held it was unconstitutional for a public school to require students to recite a prayer.
The specific challenge involved the New York Reagents' Prayer, which a group of parents protested violated their religious practices:
"Almighty God, we acknowledge our dependence upon Thee, and we beg Thy blessings upon us, our parents, our teachers and our country. Amen." In a 6-1 decision authored by Justice Hugo Black, the Court held that the prayer violated the Establishment Clause of the First Amendment because it promoted a religious belief regardless of the actual words being ambiguous as to which religion was being endorsed. The Court held that the Establishment Clause prohibited the government from endorsing a class of religions that recognized an Almighty God.
Other landmark cases banning prayer and Bible-reading in public schools include:
Abington School District v. Schempp, 374 US 203 (1963)
Chamberlin v. Dade County Board, 377 US 402 (1964)
Wallace v. Jaffre, 472 US 38 (1985)
Lee v. Weisman, 505 US 577 (1992)
Santa Fe Independent School District v. Doe, 530 US 290 (2000)
It's worth remembering, however, that no case or series of cases have "banned" prayer in schools. The Court has ruled that the government (in the person of public school teachers or other public school officials) can't endorse a religion by requiring students to pray or participate in bible study; that doesn't mean students themselves are prohibited from praying. The private exercise of religion is not prohibited in public schools.
For more information, see Related Questions, below.