It all depends on whether or not conspiracy is considered a "lesser included offense" to the actual crime. Criminal law isn't my specialty, so I don't know the answer to that question.
I think the answer to that is "it depends." If the elements that need to be proved to establish that the defendant committed the crime are also the elements that need to be proved to establish that the defendant was involved in the conspiracy, then conspiracy is a lesser included offense, and charging him with conspiracy after already trying him for the crime would amount to double jeopardy. That's the lesson of Rutledge v. US, 517 US 292
(1996) (which isn't, strictly speaking, a double jeopardy case, but it's close enough).
On the other hand, if the elements of the crime and the elements of the conspiracy are different, then conspiracy is a separate offense and there's no double jeopardy. So, for instance, in Rutledge
, a defendant was charged with conspiracy to sell drugs and running a criminal enterprise to sell drugs. The elements of those two offenses were essentially the same (colluding with others to sell drugs), so the court held that he couldn't be convicted of both. In contrast, if a defendant is charged with murder and with conspiracy to commit murder, I would guess that the elements of the two crimes are sufficiently distinct that the defendant could be convicted of both crimes.