Rapidly diminishing Arctic summer sea ice is a strong signal of the pace of global climate change. We provide point, interval, and density forecasts for four measures of Arctic sea ice: area, extent, thickness, and volume. Importantly, we enforce the joint constraint that these measures must simultaneously arrive at an ice-free Arctic. We apply this constrained joint forecast procedure to models relating sea ice to atmospheric carbon dioxide concentration and models relating sea ice directly to time. The resulting "carbon-trend" and "time-trend" projections are mutually consistent and predict a nearly ice-free summer Arctic Ocean by the mid-2030s with an 80% probability. Moreover, the carbon-trend projections show that global adoption of a lower carbon path would likely delay the arrival of a seasonally ice-free Arctic by only a few years.