Am I allowed to do a six-week internship in Germany on a 'normal' Schengen visa?

If you are staying for six weeks, what you need is in any case a regular Schengen short-stay visa. But if what you are doing count as work, you need to secure a separate “authorization” before applying for said visa. In practice, your prospective employer generally has to apply on your behalf to the Bundesagentur für Arbeit and, once they get the approval, to send you a document that you can present to the consulate. So you need to get in touch with your employer about this as soon as possible.

The German Foreign Office has a page (in German!) detailing under what conditions an internship is allowed. Some things like internships funded by government money can be done without any specific authorization but I think you would need one. You might want to ask the consulate or try to get advice from someone who knows German law better to make sure.

Whatever you do, be absolutely forthcoming in your application. Do not try to pretend you are going for tourism and then enroll in an internship, you could get in serious trouble.

Also, if you already have a multiple-entry short-stay visa, be very carefully if you are considering using it for your internship. You want to make absolutely sure that it is allowed by the rules and that your employer completed all required formalities. Definitely have the authorisation from the Federal Employment Agency (Bundesagentur für Arbeit) with you when crossing the border.