diff --git a/langchain/callbacks/streaming_aiter.py b/langchain/callbacks/streaming_aiter.py
index 45a4459b..47372dd9 100644
--- a/langchain/callbacks/streaming_aiter.py
+++ b/langchain/callbacks/streaming_aiter.py
@@ -47,14 +47,19 @@ class AsyncIteratorCallbackHandler(AsyncCallbackHandler):
         while not self.queue.empty() or not self.done.is_set():
             # Wait for the next token in the queue,
             # but stop waiting if the done event is set
-            done, _ = await asyncio.wait(
+            done, other = await asyncio.wait(
                 [
+                    # NOTE: If you add other tasks here, update the code below,
+                    # which assumes each set has exactly one task each
                     asyncio.ensure_future(self.queue.get()),
                     asyncio.ensure_future(self.done.wait()),
                 ],
                 return_when=asyncio.FIRST_COMPLETED,
             )
 
+            # Cancel the other task
+            other.pop().cancel()
+
             # Extract the value of the first completed task
             token_or_done = cast(Union[str, Literal[True]], done.pop().result())