Here's my translation of the letter they sent them. Anything in 's are my own additions.
Participation in copyright infringement
The Rights Alliance represent among others Noble Entertainment, Nordic Film and SF [Swedish Film], companies which hold the copyright to many Swedish movies that without permission are made available to the public through the Pirate Bay. The Pirate Party and Serious Tubes provide internet access to the Pirate Bay.
In February 2012 the Supreme Court decided to not allow an appeal in the so called Pirate Bay case. Through that decision it was established that not only those who administrate an illegal file sharing service but also those that provide internet access to such an illegal service make themselves guilty of a crime. Despite the fact that the Court has made this clear you continue to do so [provide the internet access]. Since all legal avenues have been exhausted, the only thing left for us to do is request that you cease providing internet access to the Pirate Bay
Through this letter you are formally informed that infringement of rights that are protected by copy right law are occurring through the Pirate Bay and that you participate in these infringements by providing internet access to the Pirate Bay.
Copyright law provides that an injunction can be imposed on those that participate in copyright infringement, fines being the consequence of non-compliance. Liability for damages for any harm caused is also established. These rules apply to among others juridical persons [companies], including non-profit/charitable organisations like the Pirate Bay and Serious Tubes, their board of directors and other associates.
According to copy right law, participating in copyright infringement is extensive. Four people have been handed prison terms and significant damages for participation in copyright infringement for, inter alia [among other things], participation in the running of and the supply of broadband internet/internet access to the Pirate Bay. Furthermore, Svea Hovratt [the Court of Appeal, inferior only to the Supreme Court] have forbidden two companies with threats of fines of 500000 SEK (US$ 80000; £52000) from in the first case supplying internet access to Pirate Bay and in the second case from providing internet access to a tracker that was used for illegal file sharing (Svea Hovratt Case No. Ö 7131-09, Case No. Ö 8873-09 and Case No. Ö 10146-09. Furthermore, Stockholm District Court established by judgment of the 16th of October 2012 the injunctions established by Svea Hovratt in Svea Hovratt Case No’s Ö 7131-09 and 8873-09 (Stockholm District Court Case No. T 7540-09 and Case No. T 11712-09.
With this as a background the copyright holders that we represent request that the Pirate Party and Serious Tubes immediately cease supplying internet access to the Pirate Bay. If notification of having done so has not been received by latest the 26th of February 2013 the Rights Alliance will pursue legal action against you without further communications.
If you would like more information you are welcome to contact us.
The Rights Alliance.
They're using scare tactics. They are referring to cases and trying to make it look like those cases have established that what they are doing is illegal by analogy, but I seriously doubt any Court would entertain that notion and I would expect that they would try any case fully on its merits. I think that becomes even more apparent when they threaten to go after the board of directors personally and threaten that they can be held liable for damages for the copyright infringement in personam [out of their own pockets].