Will Dart scripts run natively in the browser?

This is speculation, but I guess that Chrome will likely support dart natively in the future. It will mean that Chrome will be able to outperform other browsers that don't have the native support.

Update: There's a hint from The Official Google Code Blog that it might be coming:

The Dart VM is not currently integrated in Chrome but we plan to explore this option.

Update 2: It looks like I was right! Here's some recent news:

Additionally, and perhaps most importantly, Google is releasing Dartium, a Chromium build with native Dart support.

It’s more than likely that Dartium is part of a series of steps that Google will be taking to launch native support in Chrome — greatly improving the usefulness of Dart. Bringing native support into other browsers like Firefox and IE, on the other hand, will be quite the challenge.

Update 3: Since I posted this answer, Microsoft have released TypeScript which is a competing technology. I'd say it's unlikely that Microsoft have any intention of integrating Dart into Internet Explorer as they would probably prefer people to use TypeScript rather than Dart.

The official answer is Yes, Dart will run natively in Chrome. The branch of Chrome, affectionately called Dartium, is now available as source, and binaries will be released soon. More information on how to get and build the source: http://code.google.com/p/dart/wiki/BuildingDartium

As of March 25 of 2015, the offical answer is: no.

Source: Dart for the Entire Web

Google dropped the integration plan of the Dart VM into Chrome in favor of optimizing the Javascript compilation. Dartium is still accessible for development purposes.