Most of the other text on this page was written for the 2012 version, though I've made some minor changes to update for 2013. I might start doing one every year and maybe for the June solstice as well.
Accurate to the second
As far as I know this is the only astronomically accurate 2013 solstice countdown timer on the web that is correct to the nearest second.
Calculating the December 2012 Solstice
Since I couldn't find it anywhere else on the web, I used online NASA computers to calculate the exact time of the 21 December 2012 solstice, accurate to the second. Unfortunately I only thought of doing this on the morning of Friday 21 December 2012 — since I had assumed that it would have already been done to death. But to my great surprise, I couldn't even find a site with the accurate time of the solstice, let alone a countdown timer that was correct. I wish I'd done it several weeks (or months) before, as I would have got millions of hits, lol. 13 hours left wasn't enough time for google (or anyone really) to even find that page.
The clock on december212012.com was wrong by an hour and 37.627 seconds (unless they fixed it in the final few hours after I looked — I was writing this when there was 12 hours left on it and there should have been 13). As you could have easily veriied by an internet search for "time in GMT". My 2012 clock was wrong by about 1/3 of a second, because I couldn't find a flash countdown timer that allowed me to enter the end time in fractions of a second (and I didn't have the time to program one in the remaining 13 hours before the solstice, when there was important things like Christmas shopping to do today).
All the websites I could find gave the time as either 11:11 or 11:12 UTC (or GMT which is basically the same thing), i.e. Greenwich Mean Time, otherwise known as Coordinated Universal Time.
Which suggested strongly that the exact time of the 2012 solstice was between 11:11 and 11:12. But that was all I could find, apart from some countdown style calculators, which did not impress me as being created with a huge amount of detailed astronomical background knowledge (and, after having done the calculations, turned out to be wrong).
So to find the answer it was necessary to use the NASA website and do some calculations.
Using the NASA computer-generated information, the exact time of the 2012 solstice was 11:11:37.627 UTC, i.e. 11:11 and 37.627 seconds. Which was between 11:11 and 11:12 as expected.
In Sydney time (AEDST - Australian Eastern Daylight Saving Time), which is GMT+11 hours, this is 22:11 and 37.627 seconds, or in 12 hour time, 10:11 and 37.627 seconds PM.
The December 2013 Solstice
The exact time of the December 2013 solstice is 21 December 2013 at 17:11 and 1.477 seconds. That is, 5:11 P.M in UTC (Greenwich Time).
In Sydney time (AEDST - Australian Eastern Daylight Saving Time), which is GMT+11 hours, this is 22 December 2013 at 04:11 and 1.477 seconds, or in 12 hour time, 4:11 and 1.477 seconds AM.
This countdown timer is wrong by about half a second, since the timer script only accepts whole seconds for the finish time. Details of how to do the time calculation are shown below.
What Is the Solstice?
The solstices are the moments when the position of the sun in the sky is at its northerly and southerly extremes. There are two every year, one in winter and one in summer. The December solstice is summer in the Southern hemisphere and winter in the Northern hemisphere, and it is the moment in time when the sun is at its most southerly position in the sky. This is an exact moment in time (just like how noon, when the sun is directly overhead, is an exact moment in time). Once the December solstice has passed, the days get shorter (in the Southern hemisphere) and longer (in the Northern hemisphere). There are also two equinoxes, one in spring and one in autumn, when the sun is exactly in line with the Earth's equator and the day is the same length as the night.
Do an internet search for "solstice" if you would like to find out more information about this.
Advanced Background Information
The "ecliptic" is the line representing the Earth's orbit around the sun projected onto the sky. This is the same as saying that it is the line projected onto the sky where the sun is always found. The sun travels one full circle around the ecliptic once a year. The "ecliptic longitude" (used in the explanation below) is a coordinate system, just like the latitude and longitude on Earth, but using the ecliptic line as the reference point rather than the Earth's equator. The zero point of ecliptic longitude is defined to be what is called the First Point of Aries, which is where the sun crosses the celestial equator (that's the Earth's equator projected onto the sky) in March. This is one of the equinoxes (the other is in September when it again crosses the celestial equator on the other side of the Earth's yearly orbit). The units are degrees, so that there are 360 degrees in the full circle, and the ecliptic coordinates of 0 degrees corresponds to the March equinox, 90 degrees is the June solstice, 180 degrees is the September equinox, and 270 degrees is the December solstice.
If you are interested in these types of things, you can use a plantarium software program like the excellent "Stellarium" to display the ecliptic and see the path of the sun and experiment with how it all works.
Details of the Calculation
For people who are interested, this is how the calculation was done.
I used the web based interface to the NASA computer system called HORIZONS, "which can be used to generate ephemerides for solar-system bodies". This looks official enough.
The actual moment of the solstice is when the sun is located at exactly 270.000000 degrees of ecliptic longitude.
You can see that this occurs between 17:11 and 17:12. Using the two values of ecliptic longitude for 17:11 and 17:12 and some high school maths, the exact time when the sun is at 270.00000 degrees can be calculated.
270.0006896 - 269.9999826 = 0.0007070 degrees, which is how many degrees the sun moves in the one minute between 17:11 and 17:12.
and 270.000000 - 269.9999826 = 0.0000174 degrees, which is how many degrees the sun moves between 17:11 and the precise moment of the solstice.
The second result above as a fraction of the first (i.e. 0.0000174 / 0.0007070 = 0.024611033) gives the fraction of the minute (between 17:11 and 17:12) when the exact solstice occurs. Multiplying this by 60 gives the value in seconds, as 1.4767 seconds.