Would this diet work?

I believe you will lose weight from this diet, this is called the Soup Diet - it is a fad. However, as it is just a diet for a week you return to your regular diet and the weight will be gained back, that's not rocket science. Temporary diets = Temporary weight loss. It is a fad diet, fad diets are not worth doing. They are ultimately unhealthy for the body, result in massive water loss and can mess with your metabolism. The only real way to lose weight and keep it off is to adopt and maintain a healthy eating plan and exercise regime. Any diet that you cannot keep up in the long term is not worth doing.