You actually need a lot more than that. Most significantly, you need to have registered the work with the Copyright Office.
“No civil action for infringement of the copyright in any United States work shall be instituted until ... registration of the copyright claim has been made in accordance with this title.” 17 USC §411(a).
Copyright is granted to the creator upon creation.
If the work is unpublished for the purposes of the Copyright Act, you do have to register (or preregister) the work prior to the infringement. 17 USC § 412(1).
If the work is published, you still have to register it within the earlier of (a) three months after the first publication of the work or (b) one month after the copyright owner learns of the infringement.
See below for the actual text of the law.
Publication, for the purposes of the Copyright Act, generally means transferring or offering a copy of the work for sale or rental. But there are many cases where it’s not clear whether a work has or has not been published — most notably when a work is posted online and can be downloaded, but has not been explicitly offered for sale.
Also, the Supreme Court recently ruled that the mere filing of an application for registration is insufficient to file suit. The Register of Copyrights has to actually grant your application. The registration process typically takes many months, though you can pay $800 for expedited processing, if you need it.
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Here is the relevant portion of the Copyright Act:
In any action under this title, other than an action brought for a violation of the rights of the author under section 106A(a), an action for infringement of the copyright of a work that has been preregistered under section 408(f) before the commencement of the infringement and that has an effective date of registration not later than the earlier of 3 months after the first publication of the work or 1 month after the copyright owner has learned of the infringement, or an action instituted under section 411(c), no award of statutory damages or of attorney’s fees, as provided by sections 504 and 505, shall be made for—
(1) any infringement of copyright in an unpublished work commenced before the effective date of its registration; or
(2) any infringement of copyright commenced after first publication of the work and before the effective date of its registration, unless such registration is made within three months after the first publication of the work.