diff --git a/NEWS b/NEWS
index ff522766e..1bbb7c628 100644
--- a/NEWS
+++ b/NEWS
@@ -130,6 +130,8 @@ New in spot 1.99.6a (not yet released)
     was registered several times.
   * product() would incorrectly mark the product of two
     sttuter-sensitive automata as stutter-sensitive as well.
+  * complete() could incorrectly reuse an existing (but accepting!)
+    state as a sink.
 
 New in spot 1.99.6 (2015-12-04)
 
diff --git a/spot/twaalgos/complete.cc b/spot/twaalgos/complete.cc
index 1c6bfd9b2..cdccfde6b 100644
--- a/spot/twaalgos/complete.cc
+++ b/spot/twaalgos/complete.cc
@@ -45,27 +45,38 @@ namespace spot
       }
     else
       {
-	// Loop over the states and seek a state that has only self
-	// loops, and that is not accepting.  This will be our sink
-	// state.  Note that we do not even have to ensure that the
-	// state is complete as we will complete the whole automaton
-	// in a second pass.
+	// Loop over the states and search a state that has only self
+	// loop labeled by the same non-accepting mark.  This will be
+	// our sink state.  Note that we do not even have to ensure
+	// that the state is complete as we will complete the whole
+	// automaton in a second pass.
 	for (unsigned i = 0; i < n; ++i)
 	  {
-	    bool selfloop = true;
-	    acc_cond::mark_t accsum = 0U;
+	    bool sinkable = true;
+	    bool first = true;
+	    acc_cond::mark_t commonacc = 0U;
 	    for (auto& t: aut->out(i))
 	      {
 		if (t.dst != i)	// Not a self-loop
 		  {
-		    selfloop = false;
+		    sinkable = false;
+		    break;
+		  }
+		if (first)
+		  {
+		    commonacc = t.acc;
+		    first = false;
+		  }
+		else if (t.acc != commonacc)
+		  {
+		    sinkable = false;
 		    break;
 		  }
-		accsum |= t.acc;
 	      }
-	    if (selfloop && !aut->acc().accepting(accsum))
+	    if (sinkable && !aut->acc().accepting(commonacc))
 	      {
 		// We have found a sink!
+		um.second = commonacc;
 		sink = i;
 		break;
 	      }
@@ -74,8 +85,7 @@ namespace spot
 
     unsigned t = aut->num_edges();
 
-    // If the automaton is empty,
-    // pretend that state
+    // If the automaton is empty, pretend that state 0 is a sink.
     if (t == 0)
       sink = init;
 
diff --git a/tests/core/complete.test b/tests/core/complete.test
index 19f82465a..27a76d931 100755
--- a/tests/core/complete.test
+++ b/tests/core/complete.test
@@ -1,7 +1,7 @@
 #! /bin/sh
 # -*- coding: utf-8 -*-
-# Copyright (C) 2015 Laboratoire de Recherche et Développement de
-# l'Epita (LRDE).
+# Copyright (C) 2015, 2016 Laboratoire de Recherche et Développement
+# de l'Epita (LRDE).
 #
 # This file is part of Spot, a model checking library.
 #
@@ -57,6 +57,31 @@ State: 0
 State: 1
 [0] 1
 --END--
+/* This example comes from Alexandre Lewkowicz */
+HOA: v1
+States: 2
+Start: 0
+AP: 1 "a"
+Acceptance: 2 Fin(0)&Inf(1)
+--BODY--
+State: 0
+[0] 1
+State: 1
+[0] 1 {0}
+[!0] 1 {1}
+--END--
+HOA: v1
+States: 2
+Start: 0
+AP: 1 "a"
+Acceptance: 2 Fin(0)&Inf(1)
+--BODY--
+State: 0
+[0] 1
+State: 1
+[0] 1 {0}
+[!0] 1 {0}
+--END--
 EOF
 
 cat >expected <<EOF
@@ -112,6 +137,40 @@ State: 1
 [0] 1
 [!0] 1
 --END--
+HOA: v1
+States: 3
+Start: 0
+AP: 1 "a"
+acc-name: Rabin 1
+Acceptance: 2 Fin(0) & Inf(1)
+properties: trans-labels explicit-labels trans-acc complete
+properties: deterministic
+--BODY--
+State: 0
+[0] 1
+[!0] 2
+State: 1
+[0] 1 {0}
+[!0] 1 {1}
+State: 2
+[t] 2
+--END--
+HOA: v1
+States: 2
+Start: 0
+AP: 1 "a"
+acc-name: Rabin 1
+Acceptance: 2 Fin(0) & Inf(1)
+properties: trans-labels explicit-labels state-acc complete
+properties: deterministic
+--BODY--
+State: 0
+[0] 1
+[!0] 1
+State: 1 {0}
+[0] 1
+[!0] 1
+--END--
 EOF
 
 run 0 autfilt -CH automaton >out