This should do the trick:

npm i --save-dev karma-phantomjs-launcher

Then modify the plugins property of the karma.conf.js file, adding the PhantomJS plugin to the list. Also add PhantomJS to the browsers property.

plugins: [
require( 'karma-jasmine' ),
require( 'karma-chrome-launcher' ),
require( 'karma-phantomjs-launcher' ),
require( 'karma-remap-istanbul' ),
require( 'angular-cli/plugins/karma' )
],
...
browsers: [ 'PhantomJS', 'Chrome' ],

Since you want a completely headless experience, you can remove Chrome from the browsers property, and remove the karma-chrome-launcher from the plugins array as well.

As a more complete answer based on William Hampshire's one, Cuga's comment and my personal additions.

Short answer: using ChromeHeadless

You can just use Headless Chrome:

ng test --browsers ChromeHeadless

You need to have Chrome 59+.

But if you need PhantomJS (and/or chaning the default ng test behaviour with no arguments) read the following.

Longer answer: using PhantomJS

EDIT: Be aware that PhantomJS project has been archived, see this thread.

Setup

In order to be able to (optionally) run your tests without a browser, using PhantomJS, you should:

1) Install some dependencies:

npm install --save-dev karma-phantomjs-launcher
npm install --save intl

2) Add PhantomJS to the Karma's plugin list

Open karma.conf.js and add require('karma-phantomjs-launcher') to the plugins array, for example:

module.exports = function (config) {
config.set({
basePath: '',
frameworks: ['jasmine', '@angular/cli'],
plugins: [
require('karma-jasmine'),
require('karma-chrome-launcher'),
require('karma-phantomjs-launcher'),
// ...
],

3) Enable polyfills

Open your src/polyfills.ts file and uncomment the following lines:

BROWSER POLYFILLS

import 'core-js/es6/symbol';
import 'core-js/es6/object';
import 'core-js/es6/function';
import 'core-js/es6/parse-int';
import 'core-js/es6/parse-float';
import 'core-js/es6/number';
import 'core-js/es6/math';
import 'core-js/es6/string';
import 'core-js/es6/date';
import 'core-js/es6/array';
import 'core-js/es6/regexp';
import 'core-js/es6/map';
import 'core-js/es6/weak-map';
import 'core-js/es6/set';

APPLICATION IMPORTS

import 'intl';
import 'intl/locale-data/jsonp/en';

How to run the tests

Specifying the browsers when running the command

No you can either run the test using Chrome (the angular-cli default):

ng test --browsers Chrome

Or PhantomJS (headless):

ng test --browsers PhantomJS

Changing the default behaviour of just ng test

It is possible to change the default behaviour of ng test (so when no --browsers argument is provided) by changing the value of the browsers array in karma.conf.js.

It can now be set to just use Chrome (default angular-cli setup):

browsers: ['Chrome'],

or PhantomJS:

browsers: ['PhantomJS'],

or even both:

browsers: ['Chrome', 'PhantomJS'],

Angular provides all what we need to run tests without a browser.

Open the karma.conf.js file and add the ChromeHeadless to the browsers array -

browsers: ['ChromeHeadless'],

Happy testing!

If the purpose of running this headlessly is to get your code to run on a CI server, you will also likely want to set singleRun: true in your karma.conf.js. Without this Karma will run continuously, and time out.

Working Solution on Angular 11.

karma.conf.js

   browsers: ["ChromeHeadless"],
singleRun: true

And then just run ng test or ng test --project=project_name.

This works in a CI environment too.

Note: This is a combination of two different answers here.