Birmingham councillor Gareth Compton posted a message after hearing journalist Yasmin Alibhai-Brown claim on the radio that Prime Minister David Cameron did not have the moral authority to chastise the Chinese government about its human rights record because of the torture allegations surrounding the behaviour of the forces which invaded Iraq.
His message said: "can someone please stone Yasmin Alibhai-Brown to death? I shan't tell Amnesty if you don't. It would be a blessing, really".
Compton was arrested for an offence under the Communications Act, according to news service the Press Association. The arrest reportedly was for breach of the section of the law which says that it is an improper use of a public electronic communications network for someone to send "by means of a public electronic communications network a message or other matter that is grossly offensive or of an indecent, obscene or menacing character".
Compton, a barrister, has been suspended from the Conservative Party.
Paul Chambers has lost an appeal against a conviction handed down earlier this year for a joke he made on Twitter in frustration at the closing of an airport he was due to fly out of because of snow.
"Crap! Robin Hood airport is closed. You've got a week and a bit to get your shit together otherwise I'm blowing the airport sky high!!" said the message.
The men in both cases claimed that their messages were jokes.
The judge in Chambers' appeal said that he should have been aware that his message would be taken seriously in the context of increased fears about airplane attacks in recent years.
"Anyone in this country in the present climate of terrorist threats, especially at airports, could not be unaware of the possible consequences," said the judge, according to The Guardian newspaper.
The judge said the message was "menacing in its content and obviously so. It could not be more clear. Any ordinary person reading this would see it in that way and be alarmed".
Chambers' case has become a rallying point for protesters who think that laws are being misused in a way that infringes on people's rights to free speech. Chambers's supporters have included Stephen Fry, who previously said he would pay his fine.
The message was spotted by an off-duty member of airport staff and was reportedly not treated as a serious threat to the airport by security staff at the time. But the court has ruled that it was menacing.