It is my understanding that a child born in the United States is considered a U.S. citizen. Is this the case if both parents of the child are in the United States illegally?

You are correct. The general rule is that persons born in the United States are, subject to a few narrow exceptions, U.S. citizens at birth. This doctrine, known as jus soli, is derived from British common law. Recognition of jus soli appears in the Fourteenth Amendment which states that all persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.

Despite the clear wording of the Fourteenth Amendment, controversy persisted until the matter was definitively resolved by the U.S. Supreme Court in U.S. v. Wong Kim Ark, 169 U.S. 649, 18 S. Ct. 456, 42 L. Ed. 890 (1898). The court ruled that the common law principle of jus soli had been adopted as the citizenship rule of this country at its inception and that the Fourteenth Amendment was intended to be a declaration of the pre-existing common law rule.

U.S. v. Wong Kim Ark also confirmed that the status of a person born in the United States was unaffected by the status of the person's parents, except in rare cases. Therefore, even where both parents are illegal aliens, a child born in the United States will be considered a U.S. citizen at birth.