Murray, who has become a blogger and pro-independence campaigner in recent years, was judged to have been in contempt of court over blogs he wrote during the trial of former first minister Alex Salmond.
The 62-year-old’s offending blog posts contained details which, if pieced together, could lead readers to identify women who made allegations against Mr Salmond, who was acquitted of all 13 charges including sexual assault and attempted rape in March last year.
The former diplomat arrived at St Leonard’s police station in Edinburgh on Sunday morning to hand himself in and was joined by family and supporters.
Murray drank from what appeared to be a bottle of champagne and thanked the crowd for their attendance, one video on social media showed, before embracing his wife Nadira and five-month-old son Oscar.
At a virtual sentencing in May, Lady Dorrian said Murray knew there were court orders giving the women anonymity and he was “relishing” the potential disclosure of their identities.
Lady Dorrian said Murray deliberately risked what is known as “jigsaw identification”, saying: “It appears from the posts and articles that he was in fact relishing the task he set himself, which was essentially to allow the identities of complainers to be discerned – which he thought was in the public interest – in a way which did not attract sanction.”
On Friday, the Craig Murray Justice campaign group said his conviction “sets a dangerous legal precedent for freedom of speech and equality before the law”.
Several protesters gathered outside St Leonard’s on Friday afternoon and were joined by Murray.
On Friday, Murray tweeted that “in my absence the Craig Murray Justice Campaign will be continuing the fight”, linking to a statement which suggested he would start his jail term “with a clean conscience”.
Neale Hanvey, the Alba MP for Kirkcaldy and Cowdenbeath, said on Twitter he was “devastated and sickened by this development”, adding: “Scotland is no longer a safe, tolerant or free country.
“The vindictiveness of those who wield power is on naked display.”
Last month, Murray was refused in his bid to appeal the contempt finding against him at the UK Supreme Court, in a decision made by the High Court in Edinburgh
In February, Clive Thomson, who tweeted the names of women who gave evidence against Mr Salmond at his trial, was jailed for six months.
The 52-year-old carried out a “blatant and deliberate” breach of a contempt of court order banning the identification of the complainers by naming five of them on social media, said Lady Dorrian at the High Court in Edinburgh.