Getting a menu delivered via REST
This was the correct answer:

public function get($entity) {
  $menu_name = $entity;
  $menu_parameters = \Drupal::menuTree()-&gt;getCurrentRouteMenuTreeParameters($menu_name);
  $tree = \Drupal::menuTree()-&gt;load($menu_name, $menu_parameters);
  $result = array();

  foreach ($tree as $element) {
    $link = $element-&gt;link;
    array_push($result, array(
        'title' =&gt; $link-&gt;getTitle(),
        'url' =&gt; $link-&gt;getUrlObject()-&gt;getInternalPath(),
        'weight' =&gt; $link-&gt;getWeight()
      )
    );
  }
  return new ResourceResponse(json_encode($result));
}