diff --git a/starlette/middleware/base.py b/starlette/middleware/base.py
index 2a59337e5..b49ab611f 100644
--- a/starlette/middleware/base.py
+++ b/starlette/middleware/base.py
@@ -104,6 +104,7 @@ async def __call__(self, scope: Scope, receive: Receive, send: Send) -> None:
         wrapped_receive = request.wrapped_receive
         response_sent = anyio.Event()
         app_exc: Exception | None = None
+        exception_already_raised = False
 
         async def call_next(request: Request) -> Response:
             async def receive_or_disconnect() -> Message:
@@ -150,6 +151,8 @@ async def coro() -> None:
                     message = await recv_stream.receive()
             except anyio.EndOfStream:
                 if app_exc is not None:
+                    nonlocal exception_already_raised
+                    exception_already_raised = True
                     raise app_exc
                 raise RuntimeError("No response returned.")
 
@@ -176,8 +179,7 @@ async def body_stream() -> typing.AsyncGenerator[bytes, None]:
                 await response(scope, wrapped_receive, send)
                 response_sent.set()
                 recv_stream.close()
-
-        if app_exc is not None:
+        if app_exc is not None and not exception_already_raised:
             raise app_exc
 
     async def dispatch(self, request: Request, call_next: RequestResponseEndpoint) -> Response:
diff --git a/tests/middleware/test_base.py b/tests/middleware/test_base.py
index e4e82077f..427ec44ac 100644
--- a/tests/middleware/test_base.py
+++ b/tests/middleware/test_base.py
@@ -320,6 +320,27 @@ async def passthrough(request: Request, call_next: RequestResponseEndpoint) -> R
         client.get("/")
 
 
+def test_exception_can_be_caught(test_client_factory: TestClientFactory) -> None:
+    async def error_endpoint(_: Request) -> None:
+        raise ValueError("TEST")
+
+    async def catches_error(request: Request, call_next: RequestResponseEndpoint) -> Response:
+        try:
+            return await call_next(request)
+        except ValueError as exc:
+            return PlainTextResponse(content=str(exc), status_code=400)
+
+    app = Starlette(
+        middleware=[Middleware(BaseHTTPMiddleware, dispatch=catches_error)],
+        routes=[Route("/", error_endpoint)],
+    )
+
+    client = test_client_factory(app)
+    response = client.get("/")
+    assert response.status_code == 400
+    assert response.text == "TEST"
+
+
 @pytest.mark.anyio
 async def test_do_not_block_on_background_tasks() -> None:
     response_complete = anyio.Event()