diff --git a/Lib/test/test_timeout.py b/Lib/test/test_timeout.py
index da4602f..02687b0 100644
--- a/Lib/test/test_timeout.py
+++ b/Lib/test/test_timeout.py
@@ -107,24 +107,19 @@
         self.sock.close()
 
     def testConnectTimeout(self):
-        # If we are too close to www.python.org, this test will fail.
-        # Pick a host that should be farther away.
-        if (socket.getfqdn().split('.')[-2:] == ['python', 'org'] or
-            socket.getfqdn().split('.')[-2:-1] == ['xs4all']):
-            self.addr_remote = ('tut.fi', 80)
-
-        # Lookup the IP address to avoid including the DNS lookup time
+        # Choose a private address that is unlikely to exist to prevent
+        # failures due to the connect succeeding before the timeout.
+        # Use a dotted IP address to avoid including the DNS lookup time
         # with the connect time.  This avoids failing the assertion that
         # the timeout occurred fast enough.
-        self.addr_remote = (socket.gethostbyname(self.addr_remote[0]), 80)
+        addr = ('10.0.0.0', 12345)
 
         # Test connect() timeout
         _timeout = 0.001
         self.sock.settimeout(_timeout)
 
         _t1 = time.time()
-        self.failUnlessRaises(socket.error, self.sock.connect,
-                self.addr_remote)
+        self.failUnlessRaises(socket.error, self.sock.connect, addr)
         _t2 = time.time()
 
         _delta = abs(_t1 - _t2)