The Constitution (as ratified) nowhere prohibited State secession. In fact, although the Constitution included some language/provisions lifted directly from the Articles of Confederation, the new compact very pointedly omitted any reference to "perpetual" union. Given that several ratifying States explicitly reserved the right of secession in their ratification documents, it is likely the Constitution would never have been ratified by those States, if the compact had indeed prohibited secession, or if the reservations of rights were of no effect. The constitutional amendments subsequently adopted (prior to 1860) certainly did nothing whatsoever to strengthen the argument that State secession was somehow prohibited; rather, the 10th Amendment was cited by the seceeding States in support of their formal severance of ties.
Obviously, there are different points of view; some Americans believe the Constitution (including the Bill of Rights) means "exactly what it says." Others believe in a more expansive and flexible interpretation...