Immunity as a limit on the 5th amendment
<p>I was reading over <a href="https://law.stackexchange.com/a/24817/14919">this answer</a>, and it occurred to me that, when offering immunity, the prosecutor does not have foreknowledge of what the defendant will ultimately reveal.  (After all, if the prosecutor knew the information, there would be no need for the immunity itself.)</p>

<p>So, if someone is investigated for crime A, and is offered immunity for cooperation, could the defendant then also admit to some additional, only marginally-related crime?  Or something altogether unrelated?  In practice, are strict limits on what the immunity would entail communicated prior to any admission of guilt?</p>

<blockquote>
  <p>could the defendant then also admit to some additional, only marginally-related crime?</p>
</blockquote>

<p>I suppose, but why would he or she? </p>

<blockquote>
  <p>Or something altogether unrelated?</p>
</blockquote>

<p>Again, of course that could happen, but it is not clear what motivation there would be to admit to random crimes unrelated to what the person is being questioned about. </p>

<blockquote>
  <p>are strict limits on what the immunity would entail communicated prior to any admission of guilt?</p>
</blockquote>

<p>Yep. Also, the content of what the defendant (or witness - it's not always a defendant) will say is also communicated prior to any admission. </p>

<p>In <a href="https://dealbook.nytimes.com/2012/03/29/maneuvering-for-an-immunity-deal/" rel="nofollow noreferrer">summary</a>:</p>

<blockquote>
  <p>Prosecutors can grant immunity in exchange for cooperation, or reach a
  plea bargain for a reduced sentence. Before that can happen,
  prosecutors need to know what the witness will say and how credible
  that testimony is. There is nothing worse than making a deal before
  knowing what you are getting in return.</p>
  
  <p>In order to make a deal, prosecutors and defense lawyers engage in
  what is known as a “proffer” session, in which the witness is
  interviewed and the information being offered is gauged for its
  usefulness. Once someone talks with the government, however, the Fifth
  Amendment privilege is usually lost.</p>
  
  <p>To deal with that issue, the Justice Department usually enters into an
  informal letter immunity agreement — sometimes called “Queen for a
  Day” immunity after the 1960s television game show. Such immunity is
  only good for the particular proffer session and ends as soon as the
  two sides are finished.</p>
  
  <p>Under the letter agreement, the witness provides complete information
  to the government without waiving the protections of the Fifth
  Amendment.</p>
  
  <p>Unlike immunity granted under the federal statute 18 U.S.C. § 6002,
  this is a more informal arrangement that is governed by the terms of
  the agreement worked out by the parties. Like any contract, it is
  subject to negotiation, and the leverage one party has will affect how
  much protection is afforded if there is not a resolution.</p>
  
  <p>Prosecutors typically seek to use the statement to develop new leads
  and introduce new evidence gathered as a result of the information
  provided. This is not permissible when full statutory immunity is
  granted to a witness.</p>
  
  <p>Another provision often sought by the government allows prosecutors to
  introduce the statements to impeach a defendant if the case proceeds
  to trial and the person testifies in a way that is contradicted by the
  statements made in the proffer session. This is included as a way to
  deter a witness from giving one story when seeking to resolve a case,
  and then saying something different at trial to avoid a conviction.</p>
  
  <p>The federal evidence rules usually do not permit the introduction of
  statements made during negotiations related to a potential settlement
  of a case. The letter immunity is intended to get around that by
  having the witness waive the protection so that the statements can be
  used at a trial.</p>
</blockquote>

fifth-amendment immunity