diff --git a/src/websockets/asyncio/client.py b/src/websockets/asyncio/client.py
index 63cd2be2e..684cc8b49 100644
--- a/src/websockets/asyncio/client.py
+++ b/src/websockets/asyncio/client.py
@@ -18,6 +18,8 @@
     InvalidProxyMessage,
     InvalidProxyStatus,
     InvalidStatus,
+    InvalidURI,
+    InvalidRedirectURI,
     ProxyError,
     SecurityError,
 )
@@ -493,7 +495,10 @@ def process_redirect(self, exc: Exception) -> Exception | str:
 
         old_ws_uri = parse_uri(self.uri)
         new_uri = urllib.parse.urljoin(self.uri, exc.response.headers["Location"])
-        new_ws_uri = parse_uri(new_uri)
+        try:
+            new_ws_uri = parse_uri(new_uri)
+        except InvalidURI as uri_exception:
+            raise InvalidRedirectURI(uri_exception.uri, uri_exception.msg)
 
         # If connect() received a socket, it is closed and cannot be reused.
         if self.connection_kwargs.get("sock") is not None:
diff --git a/src/websockets/exceptions.py b/src/websockets/exceptions.py
index a88deaa66..4a1fc3522 100644
--- a/src/websockets/exceptions.py
+++ b/src/websockets/exceptions.py
@@ -177,6 +177,16 @@ def __str__(self) -> str:
         return f"{self.uri} isn't a valid URI: {self.msg}"
 
 
+class InvalidRedirectURI(InvalidURI):
+    """
+    Raised when redirected to a URI that isn't a valid WebSocket URI.
+
+    """
+
+    def __str__(self) -> str:
+        return f"Redirection URI {self.uri} isn't a valid URI: {self.msg}"
+
+
 class InvalidProxy(WebSocketException):
     """
     Raised when connecting via a proxy that isn't valid.
diff --git a/tests/asyncio/test_client.py b/tests/asyncio/test_client.py
index 465ea2bdb..a12bca189 100644
--- a/tests/asyncio/test_client.py
+++ b/tests/asyncio/test_client.py
@@ -354,6 +354,25 @@ def redirect(connection, request):
             "cannot follow redirect to ws://invalid/ with a preexisting socket",
         )
 
+    async def test_not_a_websocket_redirect(self):
+        """Client raises an explicit error when redirected to an absolute URI that isn't using websocket protocole."""
+
+        def redirect(connection, request):
+            response = connection.respond(http.HTTPStatus.FOUND, "")
+            response.headers["Location"] = "https://not-a-websocket.com"
+            return response
+
+        async with serve(*args, process_request=redirect) as server:
+            host, port = get_host_port(server)
+            with self.assertRaises(InvalidURI) as raised:
+                async with connect("ws://overridden/", host=host, port=port):
+                    self.fail("did not raise")
+
+        self.assertEqual(
+            str(raised.exception),
+            "Redirection URI https://not-a-websocket.com isn't a valid URI: scheme isn't ws or wss",
+        )
+
     async def test_invalid_uri(self):
         """Client receives an invalid URI."""
         with self.assertRaises(InvalidURI):