round(0, "ermintrude") succeeded instead of producing a TypeError.  Fix this.
diff --git a/Lib/test/test_float.py b/Lib/test/test_float.py
index 573cc7e..4e918fb 100644
--- a/Lib/test/test_float.py
+++ b/Lib/test/test_float.py
@@ -367,6 +367,11 @@
             self.assertEqual(round(-INF, n), -INF)
             self.assertTrue(math.isnan(round(NAN, n)))
 
+        self.assertRaises(TypeError, round, INF, 0.0)
+        self.assertRaises(TypeError, round, -INF, 1.0)
+        self.assertRaises(TypeError, round, NAN, "ceci n'est pas un integer")
+        self.assertRaises(TypeError, round, -0.0, 1j)
+
     def test_large_n(self):
         for n in [324, 325, 400, 2**31-1, 2**31, 2**32, 2**100]:
             self.assertEqual(round(123.456, n), 123.456)
diff --git a/Python/bltinmodule.c b/Python/bltinmodule.c
index a8d8d97..80fc9b4 100644
--- a/Python/bltinmodule.c
+++ b/Python/bltinmodule.c
@@ -2130,10 +2130,6 @@
 		kwlist, &x, &o_ndigits))
 		return NULL;
 
-	/* nans, infinities and zeros round to themselves */
-	if (!Py_IS_FINITE(x) || x == 0.0)
-		return PyFloat_FromDouble(x);
-
 	if (o_ndigits == NULL) {
 		/* second argument defaults to 0 */
 		ndigits = 0;
@@ -2145,6 +2141,10 @@
 			return NULL;
 	}
 
+	/* nans, infinities and zeros round to themselves */
+	if (!Py_IS_FINITE(x) || x == 0.0)
+		return PyFloat_FromDouble(x);
+
 	/* Deal with extreme values for ndigits. For ndigits > NDIGITS_MAX, x
 	   always rounds to itself.  For ndigits < NDIGITS_MIN, x always
 	   rounds to +-0.0.  Here 0.30103 is an upper bound for log10(2). */