Fix duplicate events in live history (#113896)

homeassistant/components/history/websocket_api.py

@@ -5,7 +5,7 @@ from __future__ import annotations
 import asyncio
 from collections.abc import Callable, Iterable, MutableMapping
 from dataclasses import dataclass
-from datetime import datetime as dt
+from datetime import datetime as dt, timedelta
 import logging
 from typing import Any, cast
 
@@ -564,7 +564,10 @@ async def ws_stream(
         hass,
         connection,
         msg_id,
-        last_event_time or start_time,
+        # Add one microsecond so we are outside the window of
+        # the last event we got from the database since otherwise
+        # we could fetch the same event twice
+        (last_event_time or start_time) + timedelta(microseconds=1),
         subscriptions_setup_complete_time,
         entity_ids,
         False,  # We don't want the start time state again