Also, the last commit was 2 years ago, so I don't think it's actively maintained.
That being said, if you clone it and build it and install it for your Apache host, you can likely use Dart in Apache: https://github.com/sam-mccall/mod_dart
Easiest is to decide on an intermediate representation, such as XML or JSON.
Then write POST handlers in PHP, and POST to your Apache/PHP server from your Dart client.
Then a more specific question. Is it possible to Dart to send POST request to a server, and then accept the result at the same place in the script and not updating the page?