Though coined long before, this phrase is best remembered for its use in the 1954 Supreme Court decision regarding desegregation. Many were disappointed that implementing the court’s ruling for desegregation took so long. Supreme Court justice, Thurgood Marshall, is said to have subsequently quipped: I’ve finally figured out what "all deliberate speed" means. It means "slow."
Deliberate has many reasonably common, but highly diverse senses. Accordingly, it is generally recommended that you just learn the sense (or senses) that currently interest you.
The court said the changes should be made with all deliberate speed.
A question like the present should be disposed of without undue delay. But a State cannot be expected to move with the celerity of a private business man; it is enough if it proceeds, in the language of the English Chancery, with all deliberate speed.
Supreme Court Justice Oliver Wendell Holmes -- Virginia v. West Virginia, 222 U.S. 19–20 (1911)
While giving weight to these public and private considerations, the courts will require that the defendants make a prompt and reasonable start toward full compliance with our May 17, 1954, ruling. Once such a start has been made, the courts may find that additional time is necessary to carry out the ruling in an effective manner. ...take such proceedings and enter such orders and decrees consistent with this opinion as are necessary and proper to admit to public schools on a racially nondiscriminatory basis with all deliberate speed the parties to these cases.
Supreme Court Chief Justice Earl Warren -- Brown et al. v. Board of Education of Topeka, et al., 349 U.S. 301 (1954)