diff --git a/Lib/multiprocessing/connection.py b/Lib/multiprocessing/connection.py
index 046a7fc6009..acf43b1feff 100644
--- a/Lib/multiprocessing/connection.py
+++ b/Lib/multiprocessing/connection.py
@@ -621,16 +621,24 @@ if sys.platform == 'win32':
         def accept(self):
             self._handle_queue.append(self._new_handle())
             handle = self._handle_queue.pop(0)
-            ov = _winapi.ConnectNamedPipe(handle, overlapped=True)
             try:
-                res = _winapi.WaitForMultipleObjects([ov.event], False, INFINITE)
-            except:
-                ov.cancel()
-                _winapi.CloseHandle(handle)
-                raise
-            finally:
-                _, err = ov.GetOverlappedResult(True)
-            assert err == 0
+                ov = _winapi.ConnectNamedPipe(handle, overlapped=True)
+            except OSError as e:
+                if e.winerror != _winapi.ERROR_NO_DATA:
+                    raise
+                # ERROR_NO_DATA can occur if a client has already connected,
+                # written data and then disconnected -- see Issue 14725.
+            else:
+                try:
+                    res = _winapi.WaitForMultipleObjects(
+                        [ov.event], False, INFINITE)
+                except:
+                    ov.cancel()
+                    _winapi.CloseHandle(handle)
+                    raise
+                finally:
+                    _, err = ov.GetOverlappedResult(True)
+                    assert err == 0
             return PipeConnection(handle)
 
         @staticmethod
diff --git a/Lib/test/test_multiprocessing.py b/Lib/test/test_multiprocessing.py
index 4aa0ba8e1b8..5e471f16135 100644
--- a/Lib/test/test_multiprocessing.py
+++ b/Lib/test/test_multiprocessing.py
@@ -1878,6 +1878,22 @@ class _TestListenerClient(BaseTestCase):
             p.join()
             l.close()
 
+    def test_issue14725(self):
+        l = self.connection.Listener()
+        p = self.Process(target=self._test, args=(l.address,))
+        p.daemon = True
+        p.start()
+        time.sleep(1)
+        # On Windows the client process should by now have connected,
+        # written data and closed the pipe handle by now.  This causes
+        # ConnectNamdedPipe() to fail with ERROR_NO_DATA.  See Issue
+        # 14725.
+        conn = l.accept()
+        self.assertEqual(conn.recv(), 'hello')
+        conn.close()
+        p.join()
+        l.close()
+
 class _TestPoll(unittest.TestCase):
 
     ALLOWED_TYPES = ('processes', 'threads')
diff --git a/Modules/_winapi.c b/Modules/_winapi.c
index 6c99394b09f..af6b5d7e098 100644
--- a/Modules/_winapi.c
+++ b/Modules/_winapi.c
@@ -1287,11 +1287,9 @@ PyInit__winapi(void)
     WINAPI_CONSTANT(F_DWORD, ERROR_MORE_DATA);
     WINAPI_CONSTANT(F_DWORD, ERROR_NETNAME_DELETED);
     WINAPI_CONSTANT(F_DWORD, ERROR_NO_SYSTEM_RESOURCES);
-    WINAPI_CONSTANT(F_DWORD, ERROR_NETNAME_DELETED);
-    WINAPI_CONSTANT(F_DWORD, ERROR_NO_SYSTEM_RESOURCES);
-    WINAPI_CONSTANT(F_DWORD, ERROR_IO_PENDING);
     WINAPI_CONSTANT(F_DWORD, ERROR_MORE_DATA);
     WINAPI_CONSTANT(F_DWORD, ERROR_NETNAME_DELETED);
+    WINAPI_CONSTANT(F_DWORD, ERROR_NO_DATA);
     WINAPI_CONSTANT(F_DWORD, ERROR_NO_SYSTEM_RESOURCES);
     WINAPI_CONSTANT(F_DWORD, ERROR_OPERATION_ABORTED);
     WINAPI_CONSTANT(F_DWORD, ERROR_PIPE_BUSY);