diff --git a/rts/sm/Storage.c b/rts/sm/Storage.c
index 86bd1c2bb364b2cf3b98e6faf0a9c969bb3f2498..d002fece9c6ec0943239877aab872168fbc61f6a 100644
--- a/rts/sm/Storage.c
+++ b/rts/sm/Storage.c
@@ -686,7 +686,15 @@ StgPtr allocate (Capability *cap, W_ n)
     CCS_ALLOC(cap->r.rCCCS,n);
     
     if (n >= LARGE_OBJECT_THRESHOLD/sizeof(W_)) {
-        W_ req_blocks =  (W_)BLOCK_ROUND_UP(n*sizeof(W_)) / BLOCK_SIZE;
+        // The largest number of bytes such that
+        // the computation of req_blocks will not overflow.
+        W_ max_bytes = (HS_WORD_MAX & ~(BLOCK_SIZE-1)) / sizeof(W_);
+        W_ req_blocks;
+
+        if (n > max_bytes)
+            req_blocks = HS_WORD_MAX; // signal overflow below
+        else
+            req_blocks = (W_)BLOCK_ROUND_UP(n*sizeof(W_)) / BLOCK_SIZE;
 
         // Attempting to allocate an object larger than maxHeapSize
         // should definitely be disallowed.  (bug #1791)
diff --git a/testsuite/tests/rts/all.T b/testsuite/tests/rts/all.T
index 8b4fdfa1cc980340fe0e8fe82a9ed1f5e05a4e9e..920368a0e32f85430e9bf8c85bc2f27770445a99 100644
--- a/testsuite/tests/rts/all.T
+++ b/testsuite/tests/rts/all.T
@@ -215,3 +215,8 @@ test('T9045', [ omit_ways(['ghci']), extra_run_opts('10000 +RTS -A8k -RTS') ], c
 # I couldn't reproduce 9078 with the -threaded runtime, but could easily
 # with the non-threaded one.
 test('T9078', [ omit_ways(threaded_ways) ], compile_and_run, ['-with-rtsopts="-DS" -debug'])
+
+# 251 = RTS exit code for "out of memory"
+test('overflow1', [ exit_code(251) ], compile_and_run, [''])
+test('overflow2', [ exit_code(251) ], compile_and_run, [''])
+test('overflow3', [ exit_code(251) ], compile_and_run, [''])
diff --git a/testsuite/tests/rts/overflow1.hs b/testsuite/tests/rts/overflow1.hs
new file mode 100644
index 0000000000000000000000000000000000000000..63ed5a4e0240619f826e7e1be90ddb21578f9433
--- /dev/null
+++ b/testsuite/tests/rts/overflow1.hs
@@ -0,0 +1,11 @@
+module Main where
+
+import Data.Array.IO
+import Data.Word
+
+-- Try to overflow BLOCK_ROUND_UP in the computation of req_blocks in allocate()
+-- Here we invoke allocate() via newByteArray# and the array package.
+-- Request a number of bytes close to HS_WORD_MAX,
+-- subtracting a few words for overhead in newByteArray#.
+-- Allocate Word32s (rather than Word8s) to get around bounds-checking in array.
+main = newArray (0,maxBound `div` 4 - 10) 0 :: IO (IOUArray Word Word32)
diff --git a/testsuite/tests/rts/overflow1.stderr b/testsuite/tests/rts/overflow1.stderr
new file mode 100644
index 0000000000000000000000000000000000000000..734ca954cad581c04ea2d0357139c25edab5362e
--- /dev/null
+++ b/testsuite/tests/rts/overflow1.stderr
@@ -0,0 +1 @@
+overflow1: out of memory
diff --git a/testsuite/tests/rts/overflow2.hs b/testsuite/tests/rts/overflow2.hs
new file mode 100644
index 0000000000000000000000000000000000000000..ac72158f454a8fc282c2d27c159d6704d0788f37
--- /dev/null
+++ b/testsuite/tests/rts/overflow2.hs
@@ -0,0 +1,20 @@
+{-# LANGUAGE ForeignFunctionInterface #-}
+module Main where
+
+import Foreign
+
+-- Test allocate(), the easy way.
+data Cap = Cap
+foreign import ccall "rts_unsafeGetMyCapability" myCapability :: IO (Ptr Cap)
+foreign import ccall "allocate" allocate :: Ptr Cap -> Word -> IO (Ptr ())
+
+-- Number of words n such that n * sizeof(W_) exactly overflows a word
+-- (2^30 on a 32-bit system, 2^61 on a 64-bit system)
+overflowWordCount :: Word
+overflowWordCount = fromInteger $
+                    (fromIntegral (maxBound :: Word) + 1) `div`
+                    fromIntegral (sizeOf (undefined :: Word))
+
+main = do
+  cap <- myCapability
+  allocate cap (overflowWordCount - 1)
diff --git a/testsuite/tests/rts/overflow2.stderr b/testsuite/tests/rts/overflow2.stderr
new file mode 100644
index 0000000000000000000000000000000000000000..be65509ea92805957c6bda47d7e60deee8168418
--- /dev/null
+++ b/testsuite/tests/rts/overflow2.stderr
@@ -0,0 +1 @@
+overflow2: out of memory
diff --git a/testsuite/tests/rts/overflow3.hs b/testsuite/tests/rts/overflow3.hs
new file mode 100644
index 0000000000000000000000000000000000000000..31dfd5db53e708aaa9afb706185c117c719b60f7
--- /dev/null
+++ b/testsuite/tests/rts/overflow3.hs
@@ -0,0 +1,20 @@
+{-# LANGUAGE ForeignFunctionInterface #-}
+module Main where
+
+import Foreign
+
+-- Test allocate(), the easy way.
+data Cap = Cap
+foreign import ccall "rts_unsafeGetMyCapability" myCapability :: IO (Ptr Cap)
+foreign import ccall "allocate" allocate :: Ptr Cap -> Word -> IO (Ptr ())
+
+-- Number of words n such that n * sizeof(W_) exactly overflows a word
+-- (2^30 on a 32-bit system, 2^61 on a 64-bit system)
+overflowWordCount :: Word
+overflowWordCount = fromInteger $
+                    (fromIntegral (maxBound :: Word) + 1) `div`
+                    fromIntegral (sizeOf (undefined :: Word))
+
+main = do
+  cap <- myCapability
+  allocate cap (overflowWordCount + 1)
diff --git a/testsuite/tests/rts/overflow3.stderr b/testsuite/tests/rts/overflow3.stderr
new file mode 100644
index 0000000000000000000000000000000000000000..6c804e5048965ce02ab197ab77ddd2c17a979a3e
--- /dev/null
+++ b/testsuite/tests/rts/overflow3.stderr
@@ -0,0 +1 @@
+overflow3: out of memory