Webpack fails to parse typescript files. Module parse failed: Unexpected token

I was setting up Typescript and Webpack on an old project of mine and suddenly encountered this error:

You may need an appropriate loader to handle this file type, currently no loaders are configured to process this file. See https://webpack.js.org/concepts#loaders

Then I have created a new project from scratch that goes as follow:

webpack.config.js

const webpack = require('webpack');
const path = require('path');

module.exports = {
    mode: process.env.NODE_ENV === 'production' ? 'production' : 'development',
    entry: {
        main: './src/index.ts'
    },
    module: {
        rules: [
            {
                test: '/\.ts$',
                use: {
                    loader: 'babel-loader',
                    options: {
                        presets: [
                            '@babel/preset-env',
                            '@babel/preset-typescript'
                        ]
                    }
                },
                exclude: /node_modules/
            }
        ]
    },
    resolve: {
        extensions: ['.ts'],
    },
    plugins: [
        new webpack.CleanPlugin(),
    ],
    output: {
        filename: '[name].[contenthash].js',
        path: path.join(__dirname, 'dist')
    },

}

tsconfig.json

{
    "compilerOptions": {
        "baseUrl": "./",
        "rootDir": "src",
        "outDir": "dist",
        "lib": ["ES6", "DOM"],
        "target": "es5",
        "module": "es6",
        "noImplicitAny": true,
        "removeComments": true
    }
}

src/index.ts (sourced from here)

class Greeter {
    greeting: string;

    constructor(message: string) {
        this.greeting = message;
    }

    greet() {
        return "Hello, " + this.greeting;
    }
}

let greeter = new Greeter("world");
greeter.greet();

... and the error appears again :/

ERROR in ./src/index.ts 2:12 Module parse failed: Unexpected token (2:12) You may need an appropriate loader to handle this file type, currently no loaders are configured to process this file. See https://webpack.js.org/concepts#loaders

I feel like I'm missing something, what it can be?

ps: I have tried also different loaders as ts-loader and awesome-typescript-loader.

Thanks! Any help is appreciated :*