Deploying Cloud Run and React App from same Firebase.json file

I want to use Firebase hosting and cloud run to deploy my stateless container that runs my backend, and I want to also deploy a React App under the same hosting for my front end. Is this possible?

It seems that unless I point public to 'build' for the React app then the front end won't load, but when I do that it can no longer find the backend code?

Here is my firebase.json file:

  "hosting": {
    "public": "public",
    "ignore": [
    "rewrites": [
        "source": "/",
        "destination": "/frontend/build/index.html"
        "source": "/api/**",
        "run": {
          "serviceId": "backend",
          "region": "australia-southeast1"

My file structure is:

 -- backend
 -- frontend
    -- build

Does anyone know if it is possible to deploy a React App front end and cloud run backend from the same firebase.json hosting file?

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