On paper, it was always The Economist.
You don't find news in real-time (with the breaking bar, etc.) and it's more geared towards discovering the effect of a particular story rather than learning about the story itself for the first time.
e.g. You wouldn't find news about a shooting in a school in real-time on The Economist.. nor would you hear about foreign stories until they've had time to be analyzed and stripped down. It's pretty much The Economist paper edition in online form, and ergo it suffers much the same as traditional media on the front of news discovery.
But I would argue that it's not a coincidence that the more serious news source is a weekly. News that happens suddenly seems more likely to be fluff. Plus if anything really important happened, friends would tell you.
The biggest problem is that The Economist RSS feed is only updated once a week, so you've got this deluge of info. Still I find it my primary source for serious news. The New Yorker features some of the best investigative journalism and high-quality writing, but doesn't make an attempt to cover the full spectrum of ordinary news in any given week.
For offline news I really recommend The News Hour on PBS with Jim Lehrer. Excellent in depth coverage of important happenings worldwide. Each night they focus on 2-4 issues and speak on them for 10-20 minutes each. I learn a ton from that show. They gave that boy in a balloon story the other day about 10 seconds of coverage in the intro update before they moved on and started talking about real news.
That doesn't mean their bias didn't match some pet bias of my own, it's just that I was expecting more factual analysis and less opinion than I perceived. It was a little disappointing because they're usually so favorably reviewed.
Relating this to some of the other comments, maybe the solution is to get closer to the sources of information (disintermediate) so we can make our own decisions instead of relying on opinion-makers to make up our minds for us.