Recent and expected changes in Arctic sea ice cover, snow cover, and methane emissions from permafrost thaw are likely to result in large positive feedbacks to climate warming. There is little recognition of the significant loss in economic value that the disappearance of Arctic sea ice, snow, and permafrost will impose on humans. Here, we examine how sea ice and snow cover, as well as methane emissions due to changes in permafrost, may potentially change in the future, to year 2100, and how these changes may feed back to influence the climate. Between 2010 and 2100, the annual costs from the extra warming due to a decline in albedo related to losses of sea ice and snow, plus each year's methane emissions, cumulate to a present value cost to society ranging from US$7.5 trillion to US$91.3 trillion. The estimated range reflects uncertainty associated with (1) the extent of warming-driven positive climate feedbacks from the thawing cryosphere and (2) the expected economic damages per metric ton of CO2 equivalents that will be imposed by added warming, which depend, especially, on the choice of discount rate. The economic uncertainty is much larger than the uncertainty in possible future feedback effects. Nonetheless, the frozen Arctic provides immense services to all nations by cooling the earth's temperature: the cryosphere is an air conditioner for the planet. As the Arctic thaws, this critical, climate-stabilizing ecosystem service is being lost. This paper provides a first attempt to monetize the cost of some of those lost services.