diff --git a/h11/_connection.py b/h11/_connection.py
index e37d82a..d479bb8 100644
--- a/h11/_connection.py
+++ b/h11/_connection.py
@@ -648,8 +648,9 @@ def _clean_up_response_headers_for_sending(self, response: Response) -> Response
             # Make sure Connection: close is set
             connection = set(get_comma_header(headers, b"connection"))
             connection.discard(b"keep-alive")
-            connection.add(b"close")
-            headers = set_comma_header(headers, b"connection", sorted(connection))
+            if b"close" not in connection:
+                connection.add(b"close")
+                headers = set_comma_header(headers, b"connection", sorted(connection))
 
         return Response(
             headers=headers,
diff --git a/h11/tests/test_connection.py b/h11/tests/test_connection.py
index 01260dc..dde6765 100644
--- a/h11/tests/test_connection.py
+++ b/h11/tests/test_connection.py
@@ -1118,3 +1118,9 @@ def test_special_exceptions_for_lost_connection_in_message_body() -> None:
     with pytest.raises(RemoteProtocolError) as excinfo:
         c.next_event()
     assert "incomplete chunked read" in str(excinfo.value)
+
+
+def test_ensure_connection_close_remains_untouched() -> None:
+    c = Connection(SERVER)
+    data = c.send(Response(status_code=200, headers=[(b"connection", b"close")]))
+    assert data == b"HTTP/1.1 200 \r\n" b"connection: close\r\n\r\n"