Originally Posted by SapInMyBlood

You can definitely climb hills on a single speed it's just way less comfortable. People have done the Tour Divide on a SS

As for light bikes, there's not much off the shelf that will be under 20lbs for $1k. Used bikes however will get you there easily. Look for bikes with a 105 / Ultegra groupset, 25-28mm tires and a carbon fork if not also carbon frame. I picked up a Specialized Roubaix with ultegra groupset for $1k last year ... about 17.5 lbs

Look around, look up reviews for the bikes you find (Google + Youtube)

With that being said, 17lbs vs 23lbs isn't honestly a massive difference.