Please let us know what module you are using. Tk, GTK+, Win32::GUI, something else? In Tk you can do this by creating a tag on your Text widget using the "bind" method, to call a particular subroutine on double-click, and use the getSelected method to obtain the current selection at that point.

  my $mw = MainWindow->new;
  my $txt = $mw->Text(-height => 5, -width => 40);
    $txt->bind('<Double-Button-1>', sub { local $\ = "\n";  print $txt
+->getSelected});
    $txt->pack;
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