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If there is no assignment provision ... If the contract is silent as to assignment, you are free to assign it without permission. You should prepare an assignment agreement between the new and old entities. And you should notify the other party that the agreement has been assigned. However, your assignment won't be enforced in court if the new arrangement materially alters what’s expected under the contract -- for example, if the assignment affects the performance due under the contract, decreases the value or return anticipated, or increases the risks for the other party to the contract (the party who is not assigning contractual rights).
If there is an assignment clause ... Assignment provisions come in various forms. Typically an assignment provision prohibits any assignment of the contract. In that case you'll need permission from the other party. Many assignment clauses permit assignments under certain circumstances, most commonly if one of the parties is sold, or ownership of the company changes, or the company merges or becomes a subsidiary of another company. In that case, you may be able to assign without permission. If that's the case, you should still notify the other party that the new entity is assuming the rights and obligations under the agreement.
Don't try this at home. If the contract does not permit assignments, it is often quite easy to proceed with a transfer, anyway, particularly if both parties get along. The parties simply sign a short agreement in which they agree that one party's rights can be assigned to a third entity. You should probably seek an attorney's assistance to make sure the assignment is properly prepared.