The extension of the double jeopardy clause to the states precludes a state from trying a defendant for the same crime twice. It does not prevent two separate sovereigns from trying the defendant for the same crime. The following excerpt is from Constitution: Analysis and Interpretation pages 1490-91:[^1]

  Although the Court had long accepted in dictum the principle that prosecution
  by two governments of the same defendant for the same conduct would not
  constitute double jeopardy, it was not until United States v. Lanza[53]
  that the conviction in federal court of a person previously convicted in a
  state court for performing the same acts was sustained. “We have here two
  sovereignties, deriving power from different sources, capable of dealing with
  the same subject-matter within the same territory. . . . Each government
  in determining what shall be an offense against its peace and dignity is
  exercising its own sovereignty, not that of the other.”[54] The “dual
  sovereignty” doctrine is not only tied into the existence of two sets of laws
  often serving different federal-state purposes and the now overruled principle
  that the Double Jeopardy Clause restricts only the national government and
  not the states,[55] but it also reflects practical considerations that
  undesirable consequences could follow an overruling of the doctrine. Thus, a
  state might preempt federal authority by first prosecuting and providing for a
  lenient sentence (as compared to the possible federal sentence) or acquitting
  defendants who had the sympathy of state authorities as against federal law
  enforcement.[56] The application of the clause to the states has therefore
  worked no change in the “dual sovereign” doctrine.

  53 260 U.S. 377 (1922).
  54 260 U.S. at 382. See also Hebert v. Louisiana, 272 U.S. 312 (1924); Screws
  v. United States, 325 U.S. 91, 108 (1945); Jerome v. United States, 318
  U.S. 101 (1943).
  55 Benton v. Maryland, 395 U.S. 784 (1969), extended the clause to the states.
  56 Reaffirmation of the doctrine against double jeopardy claims as to the
  Federal Government and against due process claims as to the states occurred
  in Abbate v. United States, 359 U.S. 187 (1959), and Bartkus v. Illinois,
  359 U.S. 121 (1959), both cases containing extensive discussion and policy
  analyses. The Justice Department follows a policy of generally not duplicating
  a state prosecution brought and carried out in good faith, see Petite
  v. United States, 361 U.S. 529, 531 (1960); Rinaldi v. United States, 434
  U.S. 22 (1977), and several provisions of federal law forbid a federal
  prosecution following a state prosecution. E.g., 18 U.S.C. §§ 659, 660, 1992,
  2117. The Brown Commission recommended a general statute to this effect,
  preserving discretion in federal authorities to proceed upon certification by
  the Attorney General that a United States interest would be unduly harmed if
  there were no federal prosecution. NATIONAL COMMISSION ON REFORM OF FEDERAL
  CRIMINAL LAWS, FINAL REPORT 707 (1971).

[^1]: http://www.gpo.gov/fdsys/pkg/GPO-CONAN-2013/content-detail.h...