No, the GNU GPL 3 does not prohibit DRM.
Note that in legal terms, DRM is usually referred to as (Effective) Technological Measures (ETM).
The relvant text is in part 3 of GPLv3:
When you convey a covered work, you waive any legal power to forbid circumvention of technological measures to the extent such circumvention is effected by exercising rights under this License with respect to the covered work, and you disclaim any intention to limit operation or modification of the work as a means of enforcing, against the work's users, your or third parties' legal rights to forbid circumvention of technological measures.
This text does not forbid DRM/ETM in free software, but provides an explicit permission to circumvent, thereby removing the legal protection otherwise given to ETM by the DCMA and similar laws.
This is a compromise text resulting from a massive resistance from hardware manufacturers and some software developers (including Linus Torvalds) to initial attempts to declare DRM/ETM incompatible with GPLv3.
What if the DRM scheme requires some kind of secret to work? Wouldn't the GPL require this secret to be published and GPL licensed too, making the DRM useless?
No, it would not. DRM/ETM schemes that leverages on Public Key Encryption (PKE) technologies exists. The DRM software would be open source. However, the secret key to "unlock" the application would not, and could be delivered to individual users (presumably after they've paid a premium) out of band. The TOS for the premium service could disallow public sharing of secret keys, and/or the PKE would tie each secret key to a single device, making sharing of secret keys ineffective.
What the GPLv3 anti-DRM clause does, is to permit brute force and other cryptographic attacks on this scheme, in jurisdictions (which is most of them in 2015) that has outlawed "circumvention of technological measures".