Option to mount SSE server to existing ASGI server

Fixes modelcontextprotocol#311

Add option to mount SSE server to an existing ASGI server.

* Add a new method `sse_app` in `src/mcp/server/fastmcp/server.py` to return an instance of the SSE server app.
* Update the `run_sse_async` method in `src/mcp/server/fastmcp/server.py` to use the new `sse_app` method.
* Update the documentation in `README.md` to include instructions on how to mount the SSE server to an existing ASGI server.
* Fix the example in `README.md` to use `app.mount('/', mcp.sse_app())` instead.

---

For more details, open the [Copilot Workspace session](https://copilot-workspace.githubnext.com/modelcontextprotocol/python-sdk/issues/311?shareId=XXXX-XXXX-XXXX-XXXX).

Author: comfuture

Diff for: README.md

@@ -31,6 +31,7 @@
   - [Development Mode](#development-mode)
   - [Claude Desktop Integration](#claude-desktop-integration)
   - [Direct Execution](#direct-execution)
+  - [Mounting to an Existing ASGI Server](#mounting-to-an-existing-asgi-server)
 - [Examples](#examples)
   - [Echo Server](#echo-server)
   - [SQLite Explorer](#sqlite-explorer)
@@ -346,6 +347,21 @@ python server.py
 mcp run server.py
 ```
 
+### Mounting to an Existing ASGI Server
+
+You can mount the SSE server to an existing ASGI server using the `sse_app` method. This allows you to integrate the SSE server with other ASGI applications.
+
+```python
+from fastapi import FastAPI
+from mcp.server.fastmcp import FastMCP
+
+app = FastAPI()
+mcp = FastMCP("My App")
+
+# Mount the SSE server to the existing ASGI server
+app.mount('/', mcp.sse_app())
+```
+
 ## Examples
 
 ### Echo Server

Diff for: src/mcp/server/fastmcp/server.py

@@ -464,6 +464,22 @@ async def run_sse_async(self) -> None:
         from starlette.applications import Starlette
         from starlette.routing import Mount, Route
 
+        starlette_app = self.sse_app()
+
+        config = uvicorn.Config(
+            starlette_app,
+            host=self.settings.host,
+            port=self.settings.port,
+            log_level=self.settings.log_level.lower(),
+        )
+        server = uvicorn.Server(config)
+        await server.serve()
+
+    def sse_app(self) -> Starlette:
+        """Return an instance of the SSE server app."""
+        from starlette.applications import Starlette
+        from starlette.routing import Mount, Route
+
         sse = SseServerTransport("/messages/")
 
         async def handle_sse(request):
@@ -476,23 +492,14 @@ async def handle_sse(request):
                     self._mcp_server.create_initialization_options(),
                 )
 
-        starlette_app = Starlette(
+        return Starlette(
             debug=self.settings.debug,
             routes=[
                 Route("/sse", endpoint=handle_sse),
                 Mount("/messages/", app=sse.handle_post_message),
             ],
         )
 
-        config = uvicorn.Config(
-            starlette_app,
-            host=self.settings.host,
-            port=self.settings.port,
-            log_level=self.settings.log_level.lower(),
-        )
-        server = uvicorn.Server(config)
-        await server.serve()
-
     async def list_prompts(self) -> list[MCPPrompt]:
         """List all available prompts."""
         prompts = self._prompt_manager.list_prompts()