The internal heat of the Earth would continue to be the same, so even if the top of the ocean froze, there would be a water layer underneath it wherever the ocean floor is being created by ocean floor spreading.
I think eventually the top layer of the crust might get so cold that tectonics will slow down and even stop; as the crust sinks into the Earth at subduction zones it will cool, and I imagine the subduction zones will gradually seize up, perhaps becoming new wide mountain ranges.
But there is another source of energy which will stop the Earth becoming a frozen, stable world;
the Moon will still be there, stretching the Earth as it moves in its elliptical orbit; I think this will mean that there will always be a liquid water layer underneath the ice even when the atmosphere freezes.
And freeze it will; the oxygen won’t find very much on the surface of the Earth to oxidise (after all our entire planet is covered in oxygen now; anything that isn’t oxidised and can be doesn’t last long… if it came to a race between oxidation and freezing I think the cold would win).
In the pockets of salty water around the geothermal vents it might even be possible for life to continue;
similarly there is life in the deep warm layers of the Earth, and these layers will still be there, just a little deeper perhaps.
SF worldbuilding at
http://www.orionsarm.com/main.html
