Is there any solid evidence that any country's government has been engaged in and has been proven to have farmed human organs for sale on the black market?
There was a huge controversy towards sweden in 2009 when Aftonbladet one of our newspapers published that the israeli military had at some point been harvesting organs of dead palestinians. At the time I personally didn't believe Aftonbladets claims as virtually all Swedish newspapers are about as good as tabloids, and it sounded like something out of the Daily Mail.
21th of August 2009 the Local reports in Israel blasts Sweden over organ harvesting report calling for the Swedish government to condemn one of it's newspapers reports. Something which is nearly unthinkable in Sweden as the newspapers are completely separate entities:
Israel has announced that it intends to lodge an official protest to Sweden over its refusal to condemn an article in a national newspaper which claimed Israeli soldiers harvest the organs of dead Palestinians.
21th of December 2009 the Local ran another article when this whole mess unraveled (to my surprise) titled Israeli military admits to organ harvesting:
Following diplomatic tensions over an August article published in Swedish tabloid Aftonbladet accusing the Israeli army of illegally harvesting the organs of Palestinians, Israel has admitted its forensic pathologists removed organs from dead bodies without consent from their families, reports the Associated Press.
Another article in the Guardian goes on to explain that the incidents happened in the 90's, and that the doctor responsible was removed from his post in 2004. As opposed to what was reported in Aftonbladet the organ-harvesting program wasn't targeting Palestinians.
Israel has admitted pathologists harvested organs from dead Palestinians, and others, without the consent of their families – a practice it said ended in the 1990s – it emerged at the weekend.
The admission, by the former head of the country's forensic institute, followed a furious row prompted by a Swedish newspaper reporting that Israel was killing Palestinians in order to use their organs – a charge that Israel denied and called "antisemitic".
However according to the interviewer that discovered this, the organs weren't sold on the black market, instead they were taken to public hospitals as well as was sent to be used by the Israeli military, as reported by Al Jazeera:
Nancy Scheper-Hughes, who conducted the interview, told Al Jazeera on Monday that Hiss had said the "body parts were used by hospitals for transplant purposes - cornea transplants. They were sent to public hospitals [for use on citizens].
So there's at least one incident where one country was performing organ harvesting that they've since admitted too. Now since Israel didn't sell the organs on the black market, this answer doesn't cover your entire question, but it seemed to me that the important part of this question was on non-consensual organ harvesting.
According to this article in The Weekly Standard, Chinese prisoners are or were routinely killed for their organs. These organs are then used for medical procedures on other, presumably 'more important', people.
The article is chilling reading, and its sources appear to include Chinese doctors, officials and prisoners. According to the article, sometimes organs were harvested from living prisoners without anaesthetic.
Caveats: Wikipedia lists the Weekly Standard as a "neoconservative opinion magazine" owned by News Corp which "has never been profitable", and quotes others describing it as "redoubt of neoconservatism" and as "the neo-con bible".
I don't have any personal knowledge of the magazine's journalistic reliability; I think this article is the first and so far only one of theirs I've read. That said, I hope political bias affects opinions, not facts, even those upon which opinions are based, and the article presents its content as researched fact.
This BBC article, China announces end date for taking prisoners' organs says,
A senior Chinese official has said the country will phase out the practice of taking organs from executed prisoners from November.
In the first few years of the century, officials were still refusing to confirm reports that they used organs taken from convicted criminals for transplant operations.
But in 2006, an undercover BBC team revealed that prisoners' organs were being sold to wealthy foreign patients. China then began admitting it was, after all, using the organs.