the mpemba hypothesis suggests that a vessel of hot water placed in a freezer will quickly establish convection currents, these movements in the water prevent the surface from freezing over, the currents continue to flow, and the heat energy of the water is quickly conducted away to the air above the water and through the walls of the vessel.
a vessel of cold water will not develop the same convection currents, so the surface of the water can freeze over, acting as an insulating layer.
there has been a little research into this, and it seems there may be something in it, but conditions have to be 'just so'…
temperature gradient has a lot to do with the case described by billy; the cold water will increase in temperature at a faster rate, because there is a bigger difference driving the energy through the sides bottle, but i don't see why it should warm up to Xdegrees before the warmer water.
if there is ice in the bottle, then the temperature of the water will stay at zero until the ice has melted. latent heat and all that.