diff --git a/asgi_gzip/__init__.py b/asgi_gzip/__init__.py
index d199007..a3cfd6c 100644
--- a/asgi_gzip/__init__.py
+++ b/asgi_gzip/__init__.py
@@ -30,6 +30,7 @@ def __init__(self, app, minimum_size: int, compresslevel: int = 9) -> None:
         self.send = unattached_send
         self.initial_message = {}
         self.started = False
+        self.content_encoding_set = False
         self.gzip_buffer = io.BytesIO()
         self.gzip_file = gzip.GzipFile(
             mode="wb", fileobj=self.gzip_buffer, compresslevel=compresslevel
@@ -45,6 +46,13 @@ async def send_with_gzip(self, message) -> None:
             # Don't send the initial message until we've determined how to
             # modify the outgoing headers correctly.
             self.initial_message = message
+            headers = Headers(raw=self.initial_message["headers"])
+            self.content_encoding_set = "content-encoding" in headers
+        elif message_type == "http.response.body" and self.content_encoding_set:
+            if not self.started:
+                self.started = True
+                await self.send(self.initial_message)
+            await self.send(message)
         elif message_type == "http.response.body" and not self.started:
             self.started = True
             body = message.get("body", b"")
diff --git a/tests/test_asgi_gzip.py b/tests/test_asgi_gzip.py
index a44c480..66d62a1 100644
--- a/tests/test_asgi_gzip.py
+++ b/tests/test_asgi_gzip.py
@@ -76,3 +76,27 @@ async def generator(bytes, count):
     assert response.text == "x" * 4000
     assert response.headers["Content-Encoding"] == "gzip"
     assert "Content-Length" not in response.headers
+
+
+def test_gzip_ignored_for_responses_with_encoding_set(test_client_factory):
+    def homepage(request):
+        async def generator(bytes, count):
+            for index in range(count):
+                yield bytes
+
+        streaming = generator(bytes=b"x" * 400, count=10)
+        return StreamingResponse(
+            streaming, status_code=200, headers={"Content-Encoding": "br"}
+        )
+
+    app = Starlette(
+        routes=[Route("/", endpoint=homepage)],
+        middleware=[Middleware(GZipMiddleware)],
+    )
+
+    client = test_client_factory(app)
+    response = client.get("/", headers={"accept-encoding": "gzip, br"})
+    assert response.status_code == 200
+    assert response.text == "x" * 4000
+    assert response.headers["Content-Encoding"] == "br"
+    assert "Content-Length" not in response.headers