diff -r 5f8c68281d18 Lib/random.py
--- a/Lib/random.py	Wed Feb 06 10:37:19 2013 +0200
+++ b/Lib/random.py	Wed Feb 06 14:43:04 2013 +0200
@@ -343,7 +343,12 @@
 
         """
         u = self.random()
-        c = 0.5 if mode is None else (mode - low) / (high - low)
+        if mode is None:
+            c = 0.5
+        elif mode == high:
+            c = 1
+        else:
+            c = (mode - low) / (high - low)
         if u > c:
             u = 1.0 - u
             c = 1.0 - c
diff -r 5f8c68281d18 Lib/test/test_random.py
--- a/Lib/test/test_random.py	Wed Feb 06 10:37:19 2013 +0200
+++ b/Lib/test/test_random.py	Wed Feb 06 14:43:04 2013 +0200
@@ -512,6 +512,25 @@
             self.assertAlmostEqual(s1/N, mu, places=2)
             self.assertAlmostEqual(s2/(N-1), sigmasqrd, places=2)
 
+    def test_constant(self):
+        g = random.Random()
+        N = 100
+        for variate, args, expected in [
+                (g.uniform, (10.0, 10.0), 10.0),
+                (g.triangular, (10.0, 10.0), 10.0),
+                (g.triangular, (10.0, 10.0, 10.0), 10.0),
+                (g.expovariate, (float('inf'),), 0.0),
+                (g.gauss, (10.0, 0.0), 10.0),
+                (g.lognormvariate, (0.0, 0.0), 1.0),
+                (g.lognormvariate, (-float('inf'), 0.0), 0.0),
+                (g.normalvariate, (10.0, 0.0), 10.0),
+                (g.paretovariate, (float('inf'),), 1.0),
+                (g.weibullvariate, (10.0, float('inf')), 10.0),
+                (g.weibullvariate, (0.0, 10.0), 0.0),
+            ]:
+            for i in range(N):
+                self.assertEqual(variate(*args), expected)
+
 class TestModule(unittest.TestCase):
     def testMagicConstants(self):
         self.assertAlmostEqual(random.NV_MAGICCONST, 1.71552776992141)