How do I open a web browser (URL) from my Flutter code?

I am building a Flutter app, and I'd like to open a URL into a web browser or browser window (in response to a button tap). How can I do this?

TL;DR

This is now implemented as Plugin

 final Uri url = Uri.parse('https://flutter.dev');
 if (!await launchUrl(url)) {
      throw Exception('Could not launch $_url');
 }

Full example:

import 'package:flutter/material.dart';
import 'package:url_launcher/url_launcher.dart';

void main() {
  runApp(new Scaffold(
    body: new Center(
      child: new RaisedButton(
        onPressed: _launchURL,
        child: new Text('Show Flutter homepage'),
      ),
    ),
  ));
}

_launchURL() async {
   final Uri url = Uri.parse('https://flutter.dev');
   if (!await launchUrl(url)) {
        throw Exception('Could not launch $_url');
    }
}

In pubspec.yaml

dependencies:
  url_launcher: ^6.1.11

Check out the latest url_launcher package.

Special Characters:

If the url value contains spaces or other values that are now allowed in URLs, use

Uri.encodeFull(urlString) or Uri.encodeComponent(urlString) and pass the resulting value instead.