diff --git a/compiler/basicTypes/Demand.lhs b/compiler/basicTypes/Demand.lhs
index 6c0ae89a767f854d1a289374abc908bdd2851b56..4fbf3ca18451b714637f702927760ca6f833b242 100644
--- a/compiler/basicTypes/Demand.lhs
+++ b/compiler/basicTypes/Demand.lhs
@@ -33,6 +33,7 @@ module Demand (
         trimCPRInfo, returnsCPR, returnsCPR_maybe,
         StrictSig(..), mkStrictSig, mkClosedStrictSig, nopSig, botSig, cprProdSig, convergeSig,
         isNopSig, splitStrictSig, increaseStrictSigArity,
+        sigMayDiverge,
 
         seqDemand, seqDemandList, seqDmdType, seqStrictSig, 
 
@@ -794,6 +795,10 @@ cprProdRes _arg_tys
   | opt_CprOff = topRes
   | otherwise  = Converges $ RetProd
 
+-- Forget that something might converge for sure
+divergeDmdResult :: DmdResult -> DmdResult
+divergeDmdResult r = r `lubDmdResult` botRes
+
 vanillaCprProdRes :: Arity -> DmdResult
 vanillaCprProdRes _arity
   | opt_CprOff = topRes
@@ -1449,6 +1454,9 @@ convergeResult Diverges      = Converges NoCPR
 convergeResult (Dunno c)     = Converges c
 convergeResult (Converges c) = Converges c
 
+sigMayDiverge :: StrictSig -> StrictSig
+sigMayDiverge (StrictSig (DmdType env ds res)) = (StrictSig (DmdType env ds (divergeDmdResult res)))
+
 argsOneShots :: StrictSig -> Arity -> [[OneShotInfo]]
 argsOneShots (StrictSig (DmdType _ arg_ds _)) n_val_args
   = go arg_ds
diff --git a/compiler/stranal/DmdAnal.lhs b/compiler/stranal/DmdAnal.lhs
index 17dbb5f48d2f4517519c631a1b169ac109fdf5c9..24c627c892a4325edd07756b8db9fdedba89aa7f 100644
--- a/compiler/stranal/DmdAnal.lhs
+++ b/compiler/stranal/DmdAnal.lhs
@@ -1056,7 +1056,10 @@ updSigEnv env sigs = env { ae_sigs = sigs }
 
 extendAnalEnv :: TopLevelFlag -> AnalEnv -> Id -> StrictSig -> AnalEnv
 extendAnalEnv top_lvl env var sig
-  = env { ae_sigs = extendSigEnv top_lvl (ae_sigs env) var sig }
+  = env { ae_sigs = extendSigEnv top_lvl (ae_sigs env) var sig' }
+  where
+  sig' | isWeakLoopBreaker (idOccInfo var) = sigMayDiverge sig
+       | otherwise                         = sig
 
 extendSigEnv :: TopLevelFlag -> SigEnv -> Id -> StrictSig -> SigEnv
 extendSigEnv top_lvl sigs var sig = extendVarEnv sigs var (sig, top_lvl)