diff --git a/Lib/test/test_ssl.py b/Lib/test/test_ssl.py
index f65aceb..7edf5b2 100644
--- a/Lib/test/test_ssl.py
+++ b/Lib/test/test_ssl.py
@@ -519,6 +519,23 @@
             finally:
                 s.close()
 
+    def test_timeout_connect_ex(self):
+        # Issue #12065: on a timeout, connect_ex() should return the original
+        # errno (mimicking the behaviour of non-SSL sockets).
+        with support.transient_internet("svn.python.org"):
+            s = ssl.wrap_socket(socket.socket(socket.AF_INET),
+                                cert_reqs=ssl.CERT_REQUIRED,
+                                ca_certs=SVN_PYTHON_ORG_ROOT_CERT,
+                                do_handshake_on_connect=False)
+            try:
+                s.settimeout(0.0000001)
+                rc = s.connect_ex(('svn.python.org', 443))
+                if rc == 0:
+                    self.skipTest("svn.python.org responded too quickly")
+                self.assertIn(rc, (errno.EAGAIN, errno.EWOULDBLOCK))
+            finally:
+                s.close()
+
     def test_connect_with_context(self):
         with support.transient_internet("svn.python.org"):
             # Same as test_connect, but with a separately created context