Open the file plugins/captcha/recaptcha/recaptcha.php (make a backup first).

• At line 24, replace
  

  const RECAPTCHA_API_SERVER = "http://api.recaptcha.net";

  with

  const RECAPTCHA_API_SERVER = "http://www.google.com/recaptcha/api";

• At line 26, replace
  

  const RECAPTCHA_VERIFY_SERVER = "api-verify.recaptcha.net";
  

  with

  const RECAPTCHA_VERIFY_SERVER = "www.google.com";

• At line 118, replace
  

  $response = $this->_recaptcha_http_post(self::RECAPTCHA_VERIFY_SERVER, "/verify",
  

  with

  $response = $this->_recaptcha_http_post(self::RECAPTCHA_VERIFY_SERVER, "/recaptcha/api/verify",
  

Save the file.
This problem will be fixed in the next Joomla update.