diff --git a/CHANGELOG.md b/CHANGELOG.md
index 10933867af..bf85a404c4 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -5,6 +5,12 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## Unreleased
+
+### Added
+
+- Allow a custom message with a `Live` overflows, instead of an ellipsis. https://github.com/Textualize/rich/pull/3702
+
 ## [14.0.0] - 2025-03-30
 
 ### Added
diff --git a/CONTRIBUTORS.md b/CONTRIBUTORS.md
index 9893915a4a..d73f6918db 100644
--- a/CONTRIBUTORS.md
+++ b/CONTRIBUTORS.md
@@ -92,3 +92,4 @@ The following people have contributed to the development of Rich:
 - [chthollyphile](https://github.com/chthollyphile)
 - [Jonathan Helmus](https://github.com/jjhelmus)
 - [Brandon Capener](https://github.com/bcapener)
+- [Samuel Colvin](https://github.com/samuelcolvin)
diff --git a/rich/live.py b/rich/live.py
index 8738cf09f4..aa21314006 100644
--- a/rich/live.py
+++ b/rich/live.py
@@ -1,7 +1,19 @@
 import sys
 from threading import Event, RLock, Thread
 from types import TracebackType
-from typing import IO, Any, Callable, List, Optional, TextIO, Type, cast
+from typing import (
+    IO,
+    Any,
+    Callable,
+    List,
+    Literal,
+    Optional,
+    TextIO,
+    Tuple,
+    Type,
+    Union,
+    cast,
+)
 
 from . import get_console
 from .console import Console, ConsoleRenderable, RenderableType, RenderHook
@@ -44,7 +56,7 @@ class Live(JupyterMixin, RenderHook):
         transient (bool, optional): Clear the renderable on exit (has no effect when screen=True). Defaults to False.
         redirect_stdout (bool, optional): Enable redirection of stdout, so ``print`` may be used. Defaults to True.
         redirect_stderr (bool, optional): Enable redirection of stderr. Defaults to True.
-        vertical_overflow (VerticalOverflowMethod, optional): How to handle renderable when it is too tall for the console. Defaults to "ellipsis".
+        vertical_overflow (VerticalOverflowMethod, optional): How to handle renderable when it is too tall for the console, pass `("ellipsis", "message ...")` for a custom message on overflow. Defaults to "ellipsis".
         get_renderable (Callable[[], RenderableType], optional): Optional callable to get renderable. Defaults to None.
     """
 
@@ -59,7 +71,9 @@ def __init__(
         transient: bool = False,
         redirect_stdout: bool = True,
         redirect_stderr: bool = True,
-        vertical_overflow: VerticalOverflowMethod = "ellipsis",
+        vertical_overflow: Union[
+            VerticalOverflowMethod, Tuple[Literal["ellipsis"], str]
+        ] = "ellipsis",
         get_renderable: Optional[Callable[[], RenderableType]] = None,
     ) -> None:
         assert refresh_per_second > 0, "refresh_per_second must be > 0"
@@ -82,10 +96,18 @@ def __init__(
         self._refresh_thread: Optional[_RefreshThread] = None
         self.refresh_per_second = refresh_per_second
 
+        if isinstance(vertical_overflow, tuple):
+            vertical_overflow, vertical_overflow_message = vertical_overflow
+        else:
+            vertical_overflow_message = "..."
+
         self.vertical_overflow = vertical_overflow
+        self.vertical_overflow_message = vertical_overflow_message
         self._get_renderable = get_renderable
         self._live_render = LiveRender(
-            self.get_renderable(), vertical_overflow=vertical_overflow
+            self.get_renderable(),
+            vertical_overflow=vertical_overflow,
+            vertical_overflow_message=vertical_overflow_message,
         )
 
     @property
diff --git a/rich/live_render.py b/rich/live_render.py
index 4284cccc4b..286485fa33 100644
--- a/rich/live_render.py
+++ b/rich/live_render.py
@@ -30,10 +30,12 @@ def __init__(
         renderable: RenderableType,
         style: StyleType = "",
         vertical_overflow: VerticalOverflowMethod = "ellipsis",
+        vertical_overflow_message: str = "...",
     ) -> None:
         self.renderable = renderable
         self.style = style
         self.vertical_overflow = vertical_overflow
+        self.vertical_overflow_message = vertical_overflow_message
         self._shape: Optional[Tuple[int, int]] = None
 
     def set_renderable(self, renderable: RenderableType) -> None:
@@ -95,7 +97,7 @@ def __rich_console__(
             elif self.vertical_overflow == "ellipsis":
                 lines = lines[: (options.size.height - 1)]
                 overflow_text = Text(
-                    "...",
+                    self.vertical_overflow_message,
                     overflow="crop",
                     justify="center",
                     end="",
diff --git a/tests/test_live.py b/tests/test_live.py
index f037e4b8b4..7b72eccd37 100644
--- a/tests/test_live.py
+++ b/tests/test_live.py
@@ -83,6 +83,25 @@ def test_growing_display_overflow_ellipsis() -> None:
     )
 
 
+def test_growing_display_overflow_ellipsis_message() -> None:
+    console = create_capture_console(height=3)
+    console.begin_capture()
+    with Live(
+        console=console,
+        auto_refresh=False,
+        vertical_overflow=("ellipsis", "custom msg"),
+    ) as live:
+        display = ""
+        for step in range(5):
+            display += f"Step {step}\n"
+            live.update(display, refresh=True)
+    output = console.end_capture()
+    assert (
+        output
+        == "\x1b[?25lStep 0\n\r\x1b[2K\x1b[1A\x1b[2KStep 0\nStep 1\n\r\x1b[2K\x1b[1A\x1b[2K\x1b[1A\x1b[2KStep 0\nStep 1\n                         custom msg                         \r\x1b[2K\x1b[1A\x1b[2K\x1b[1A\x1b[2KStep 0\nStep 1\n                         custom msg                         \r\x1b[2K\x1b[1A\x1b[2K\x1b[1A\x1b[2KStep 0\nStep 1\n                         custom msg                         \r\x1b[2K\x1b[1A\x1b[2K\x1b[1A\x1b[2KStep 0\nStep 1\nStep 2\nStep 3\nStep 4\n\n\x1b[?25h"
+    )
+
+
 def test_growing_display_overflow_crop() -> None:
     console = create_capture_console(height=5)
     console.begin_capture()