Cannot find module ‘@babel/core’

I am following along with this webpack4/react tutorial:

https://www.youtube.com/watch?v=deyxI-6C2u4

I have followed it exactly up until the part where he runs npm start. The difference is, his app runs, and mine gets the error:

Cannot find module ‘@babel/core’

The full error:

ERROR in ./src/index.js
Module build failed (from ./node_modules/babel-loader/lib/index.js):
Error: Cannot find module '@babel/core'
    at Function.Module._resolveFilename (module.js:547:15)
    at Function.Module._load (module.js:474:25)
    at Module.require (module.js:596:17)
    at require (internal/module.js:11:18)
    at Object.<anonymous> (C:\Users\joeyf\Desktop\Code\Github\webpack4-sample\node_modules\babel-loader\lib\index.js:5:15)
    at Module._compile (module.js:652:30)
    at Object.Module._extensions..js (module.js:663:10)
    at Module.load (module.js:565:32)
    at tryModuleLoad (module.js:505:12)
    at Function.Module._load (module.js:497:3)
 @ multi (webpack)-dev-server/client?http://localhost:8080 (webpack)/hot/dev-server.js ./src/index.js main[2]

I have tried to reinstall babel-core but is still not being found. Here is my package.json:

{
  "name": "webpack4-sample",
  "version": "1.0.0",
  "description": "A sample setup of Webpack4 with React and Babel",
  "main": "index.js",
  "scripts": {
    "start": "webpack-dev-server --mode development --open --hot",
    "build": "webpack --mode production"
  },
  "author": "Joey Fenny",
  "license": "ISC",
  "dependencies": {
    "babel": "^6.23.0",
    "babel-cli": "^6.26.0",
    "react": "^16.4.2",
    "react-dom": "^16.4.2"
  },
  "devDependencies": {
    "babel-core": "^7.0.0-rc.4",
    "babel-loader": "^8.0.0",
    "babel-preset-env": "^1.7.0",
    "babel-preset-react": "^6.24.1",
    "html-webpack-plugin": "^3.2.0",
    "webpack": "^4.17.1",
    "webpack-cli": "^3.1.0",
    "webpack-dev-server": "^3.1.6"
  }
}

My webpack.config.js:

const path = require('path');
const HtmlWebpackPlugin = require('html-webpack-plugin');

module.exports = {
    entry: './src/index.js',
    output: {
        path: path.join(__dirname, '/dist'),
        filename: 'index_bundle.js'
    },
    module: {
        rules: [{
            test: /\.js$/,
            exclude: path.join(__dirname, '/node_modules'),
            use: {
                loader: 'babel-loader'
            }
        }]
    },
    plugins: [
        new HtmlWebpackPlugin({
            template: './src/index.html'
        })
    ]
}

Here is a link to a git repo:

https://gitlab.com/jfny/webpack4-sample

Anyone know whats going on? Thanks.