That is not exactly accurate. There are exceptions to the rule wherein the artist or presenting organization -- not the venue -- is responsible in applying for, and acquiring, a blanket license from ASCAP and/or BMI. Do not consult with strangers on a message board for legal advice.... do the prudent thing and consult with the agencies in question.
I would also encourage you to utilize SongFile (courtesy of Harry Fox Agency), which is free and can be easily accessed online, to determine the copyright holders + publishers of each song you'll be performing. You can then crosscheck this information with the agencies to determine who owns the performance rights, as it may not always be BMI or ASCAP.
It is more complex then you might think, depending on a number of factors. I'd again encourage you to consult with the agencies or at least someone you trust in the licensing and entertainment law field. Good luck.