Will the arctic conveyor, the salt/fresh melt water driven current that drives the Gulf Stream, stop once the Arctic ice cap melts?  That depends on how fast Greenland ice melts.

"Warmer temperatures are also accelerating ice melt on the nearby Greenland ice sheet, which contains enough ice to raise sea level by seven meters (23 feet). Mass loss in Greenland more than doubled between 1996 and 2005, with loss in the southeast accelerating even further since 2004. The summer of 2007 saw a record area of ice melt on Greenland, 10 percent more than the previous maximum in 2005."

"In 2006, scientists reported that "glacial earthquakes" caused by large masses of ice moving rapidly over bedrock had doubled in frequency in Greenland over the last five years. These earthquakes are associated with meltwater from the glacier surface, which flows to the base of the ice sheet and lubricates it, causing rapid glacial movement. Positive feedback mechanisms such as this meltwater lubrication accelerate the speed with which glaciers react to warmer temperatures; ice sheets once thought to change only over millennia are now seen to be responding to warmer temperatures"

This seems to indicate they are connected.  That the warmer water at the edge of Greenland glaciers where icebergs calve and the higher rainfall from warmer arctic temps, both cause the dobins (glacial lakes) at the edge of glaciers to increase.

Eventually these lakes drain into cracks in the glacier and lubricate the sliding effect at ground level under the glacier.  The increase in the speed of the slide of the glacier connects the warming arctic sea temperature to the glacial melt.

Warmer water at the edge of the glacier causes icebergs to break off quicker too and melt out through the fijords instead of clogging them up and slowing the glacier down.

I think this in turn, dooms the Gulf Stream to end shortly after arctic ice vanishes