diff --git a/changelog.d/6499.bugfix b/changelog.d/6499.bugfix
new file mode 100644
index 0000000000..299feba0f8
--- /dev/null
+++ b/changelog.d/6499.bugfix
@@ -0,0 +1 @@
+Fix support for SQLite 3.7.
diff --git a/synapse/storage/data_stores/main/events.py b/synapse/storage/data_stores/main/events.py
index da1529f6ea..998bba1aad 100644
--- a/synapse/storage/data_stores/main/events.py
+++ b/synapse/storage/data_stores/main/events.py
@@ -1039,20 +1039,25 @@ class EventsStore(
             },
         )
 
-    @defer.inlineCallbacks
-    def _censor_redactions(self):
+    async def _censor_redactions(self):
         """Censors all redactions older than the configured period that haven't
         been censored yet.
 
         By censor we mean update the event_json table with the redacted event.
-
-        Returns:
-            Deferred
         """
 
         if self.hs.config.redaction_retention_period is None:
             return
 
+        if not (
+            await self.db.updates.has_completed_background_update(
+                "redactions_have_censored_ts_idx"
+            )
+        ):
+            # We don't want to run this until the appropriate index has been
+            # created.
+            return
+
         before_ts = self._clock.time_msec() - self.hs.config.redaction_retention_period
 
         # We fetch all redactions that:
@@ -1074,15 +1079,15 @@ class EventsStore(
             LIMIT ?
         """
 
-        rows = yield self.db.execute(
+        rows = await self.db.execute(
             "_censor_redactions_fetch", None, sql, before_ts, 100
         )
 
         updates = []
 
         for redaction_id, event_id in rows:
-            redaction_event = yield self.get_event(redaction_id, allow_none=True)
-            original_event = yield self.get_event(
+            redaction_event = await self.get_event(redaction_id, allow_none=True)
+            original_event = await self.get_event(
                 event_id, allow_rejected=True, allow_none=True
             )
 
@@ -1115,7 +1120,7 @@ class EventsStore(
                     updatevalues={"have_censored": True},
                 )
 
-        yield self.db.runInteraction("_update_censor_txn", _update_censor_txn)
+        await self.db.runInteraction("_update_censor_txn", _update_censor_txn)
 
     def _censor_event_txn(self, txn, event_id, pruned_json):
         """Censor an event by replacing its JSON in the event_json table with the
diff --git a/synapse/storage/data_stores/main/events_bg_updates.py b/synapse/storage/data_stores/main/events_bg_updates.py
index efee17b929..5177b71016 100644
--- a/synapse/storage/data_stores/main/events_bg_updates.py
+++ b/synapse/storage/data_stores/main/events_bg_updates.py
@@ -90,6 +90,14 @@ class EventsBackgroundUpdatesStore(SQLBaseStore):
             "event_store_labels", self._event_store_labels
         )
 
+        self.db.updates.register_background_index_update(
+            "redactions_have_censored_ts_idx",
+            index_name="redactions_have_censored_ts",
+            table="redactions",
+            columns=["received_ts"],
+            where_clause="NOT have_censored",
+        )
+
     @defer.inlineCallbacks
     def _background_reindex_fields_sender(self, progress, batch_size):
         target_min_stream_id = progress["target_min_stream_id_inclusive"]
diff --git a/synapse/storage/data_stores/main/schema/delta/56/redaction_censor.sql b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor.sql
index fe51b02309..ea95db0ed7 100644
--- a/synapse/storage/data_stores/main/schema/delta/56/redaction_censor.sql
+++ b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor.sql
@@ -14,4 +14,3 @@
  */
 
 ALTER TABLE redactions ADD COLUMN have_censored BOOL NOT NULL DEFAULT false;
-CREATE INDEX redactions_have_censored ON redactions(event_id) WHERE not have_censored;
diff --git a/synapse/storage/data_stores/main/schema/delta/56/redaction_censor2.sql b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor2.sql
index 77a5eca499..49ce35d794 100644
--- a/synapse/storage/data_stores/main/schema/delta/56/redaction_censor2.sql
+++ b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor2.sql
@@ -14,7 +14,9 @@
  */
 
 ALTER TABLE redactions ADD COLUMN received_ts BIGINT;
-CREATE INDEX redactions_have_censored_ts ON redactions(received_ts) WHERE not have_censored;
 
 INSERT INTO background_updates (update_name, progress_json) VALUES
   ('redactions_received_ts', '{}');
+
+INSERT INTO background_updates (update_name, progress_json) VALUES
+  ('redactions_have_censored_ts_idx', '{}');
diff --git a/synapse/storage/data_stores/main/schema/delta/56/redaction_censor4.sql b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor4.sql
new file mode 100644
index 0000000000..b7550f6f4e
--- /dev/null
+++ b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor4.sql
@@ -0,0 +1,16 @@
+/* Copyright 2019 The Matrix.org Foundation C.I.C.
+ *
+ * Licensed under the Apache License, Version 2.0 (the "License");
+ * you may not use this file except in compliance with the License.
+ * You may obtain a copy of the License at
+ *
+ *    http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+DROP INDEX IF EXISTS redactions_have_censored;