diff --git a/src/waitress/task.py b/src/waitress/task.py
index 956c0c0f..4f7f0344 100644
--- a/src/waitress/task.py
+++ b/src/waitress/task.py
@@ -180,6 +180,19 @@ def has_body(self):
             or self.status.startswith("304")
         )
 
+    def set_close_on_finish(self) -> None:
+        # if headers have not been written yet, tell the remote
+        # client we are closing the connection
+        if not self.wrote_header:
+            connection_close_header = None
+            for (headername, headerval) in self.response_headers:
+                headername = "-".join([x.capitalize() for x in headername.split("-")])
+                if headername == "Connection":
+                    connection_close_header = headerval.lower()
+            if connection_close_header is None:
+                self.response_headers.append(("Connection", "close"))
+        self.close_on_finish = True
+
     def build_response_header(self):
         version = self.version
         # Figure out whether the connection should be closed.
@@ -188,7 +201,6 @@ def build_response_header(self):
         content_length_header = None
         date_header = None
         server_header = None
-        connection_close_header = None
 
         for (headername, headerval) in self.response_headers:
             headername = "-".join([x.capitalize() for x in headername.split("-")])
@@ -205,8 +217,6 @@ def build_response_header(self):
             if headername == "Server":
                 server_header = headerval
 
-            if headername == "Connection":
-                connection_close_header = headerval.lower()
             # replace with properly capitalized version
             response_headers.append((headername, headerval))
 
@@ -218,23 +228,18 @@ def build_response_header(self):
             content_length_header = str(self.content_length)
             response_headers.append(("Content-Length", content_length_header))
 
-        def close_on_finish():
-            if connection_close_header is None:
-                response_headers.append(("Connection", "close"))
-            self.close_on_finish = True
-
         if version == "1.0":
             if connection == "keep-alive":
                 if not content_length_header:
-                    close_on_finish()
+                    self.set_close_on_finish()
                 else:
                     response_headers.append(("Connection", "Keep-Alive"))
             else:
-                close_on_finish()
+                self.set_close_on_finish()
 
         elif version == "1.1":
             if connection == "close":
-                close_on_finish()
+                self.set_close_on_finish()
 
             if not content_length_header:
                 # RFC 7230: MUST NOT send Transfer-Encoding or Content-Length
@@ -244,8 +249,8 @@ def close_on_finish():
                     response_headers.append(("Transfer-Encoding", "chunked"))
                     self.chunked_response = True
 
-                if not self.close_on_finish:
-                    close_on_finish()
+                if not self.self.set_close_on_finish:
+                    self.set_close_on_finish()
 
             # under HTTP 1.1 keep-alive is default, no need to set the header
         else:
@@ -350,11 +355,7 @@ def execute(self):
         status, headers, body = e.to_response(ident)
         self.status = status
         self.response_headers.extend(headers)
-        # We need to explicitly tell the remote client we are closing the
-        # connection, because self.close_on_finish is set, and we are going to
-        # slam the door in the clients face.
-        self.response_headers.append(("Connection", "close"))
-        self.close_on_finish = True
+        self.set_close_on_finish()
         self.content_length = len(body)
         self.write(body)
 
@@ -478,7 +479,7 @@ def start_response(status, headers, exc_info=None):
                     # close the connection so the client isn't sitting around
                     # waiting for more data when there are too few bytes
                     # to service content-length
-                    self.close_on_finish = True
+                    self.set_close_on_finish()
                     if self.request.command != "HEAD":
                         self.logger.warning(
                             "application returned too few bytes (%s) "