Flutter Checkbox not working in AlertDialog

I'm trying to add a Checkbox (or CheckboxListTile) to the content of an AlertDialog. The checkbox is shown with the correct initial value, however tapping the Checkbox doesn't check/uncheck it.

This is my (simplified) code:

class MyState extends State<MyStatefulWidget> {
    bool checked = true;

    _MyState() {
        _showDialog();
    }

    Future<Null> _showDialog() async {
        return showDialog(
            context: context,
            child: new AlertDialog(
                content: new Checkbox(
                value: checked,
                onChanged: (bool value) {
                    setState(() {
                        checked = value;
                    });
                },
            ),
            actions: <Widget>[
                new FlatButton(
                    onPressed: () {
                        Navigator.of(context).pop(checked);
                    },
                    child: new Text('OK')
                )
            ],
        );
    }
}

It works when I add the Checkbox directly to the main layout, but not inside the AlertDialog. What is going on here? How can this be solved?