First, if you simply want to make passing reference to the U.S. Constitution in an APA Style paper, you can mention it in text without a reference list entry.
Law students described a great affinity for the U.S. Constitution in their
response papers.
However, if you are using some part of the U.S. Constitution as evidence to support a point you are making in your paper, you should construct the citation using Bluebook Rule 11, which covers federal and state constitutions.
All citations of the U.S. Constitution begin with U.S. Const., followed by the article, amendment, section, and/or clause numbers as relevant. The terms article, amendment, section, and clause are always abbreviated art., amend., §, and cl., respectively. Preamble is abbreviated pmbl. Article and amendment numbers are given in Roman numerals (I, II, III); section and clause numbers are given in Arabic numerals (1, 2, 3).
The Bluebook states that for parts of the Constitution currently in force, do not include a date. If you are referring to a part of the Constitution that has been repealed or amended, include the year that the part in question was repealed or amended in parentheses.
Using Rule 11, here are example in-text citation and reference list entries. Note how similar they are:
In text: The founding fathers addressed the process by which new states
may join the union (U.S. Const. art. I, § 3).
Reference list: U.S. Const. art. I, § 3.
In text: Women gained the right to vote in 1920 (U.S. Const. amend. XIX).
Reference list: U.S. Const. amend. XIX.
In text: During prohibition, the sale of liquor was made illegal (U.S.
Const. amend. XVIII, repealed 1933).
Reference list: U.S. Const. amend. XVIII (repealed 1933).