Sorry, what I meant by "stable" there is countries that have been relatively stable for a few hundred years and seem reasonably likely to continue that integrity for at least a century or so.
Of course, predicting future stability is complete guesswork!
Even the US had a close call with a fairly nasty civil war in that time frame, and in three days the 198th anniversary of the Burning of Washington happens: http://en.wikipedia.org/wiki/Burning_of_Washington "On August 24, 1814, after defeating the Americans at the Battle of Bladensburg, a British force led by Major General Robert Ross occupied Washington, D.C. and set fire to many public buildings. The facilities of the U.S. government, including the White House and U.S. Capitol, were largely destroyed."
The Thai king is the longest-reigning current head of state, ascending the throne on 9 June 1946. Elizabeth II of England is 2nd, 6 February 1952.
The oldest country (not government) is likely Vietnam (2897 BCE). Other contenders: Japan (660 BCE), China (221 BCE), Ethiopia (~800 BCE), or Iran (678 BCE).
Few of today's modern states pre-date the 19th Century, many antedate World War II or the great de-colonialisation of the 1960s including much of Africa and Oceana (some of the longest inhabited regions of Earth).
Among the more long-lived institutions are the Catholic Church (traditionally founded by Jesus ~30 AD, emerging as an institutional power in 2nd Century Rome). The oldest company I can find is Kongo Gumi, founded in 578, a Japanese construction firm. The record however is likely held by the Shishi Middle School founded in China between 143 and 141 BCE.
My own suggestion would be the Krell, though some might disqualify this based on a requirement for human organization.
That was my thought as well. However it spent a great deal of time under foreign rule: under the Greeks and Romans, the Turkish / Ottoman empire, and later under British occupation. And, I just discovered what boxer Muhammad Ali's referent was.
We are straying a bit off topic here. I don't think any country has been stable for a "few" hundred years. A few is 3. This is before the US was founded.
I would say a stable country is one which has had a legit democracy for 70 or so years and doesn't share a border with a non-democratic / non-legit-democratic state. These two points suggests its unlikely to have a revolution or be invaded any time soon.
Of course, if you look at the UK governments track record with IT.. you wouldn't trust it. I would say the same with the US, especially in concern with data security.
I think I could argue that England has been pretty stable since at least the end of the Civil War - which is 360 years ago. A lot of the institutions that form part of the current UK go back an awful lot further than that.
I'll reject your democratic requirement out of hand, as democracies haven't proven particularly stable. There was that trial-run in Athens which lasted 501 years (508 - 7 BCE, with interruptions). Other than a few small/outlying instances (most notably the Althing in Iceland, it didn't re-emerge until the short-lived Corsican Republic (1755), and of course, the United States (1776).
Japan (660 BCE) and China (221 BCE) have both had feudal / bureaucratic governments exhibiting very high levels of stability. While dynasties and eras are marked, the overall states persisted largely intact.
Of course, predicting future stability is complete guesswork!