I saw this image on Facebook and I usually take what's posted there with a grain of salt. I have reversed image searched and googled, but with no luck. I'm curious about the authenticity surrounding this letter.
As pointed out by tohuwawohu in a comment to the question (later moved to chat), the University of Bern has publicly rebutted this as a forgery, and not even a very good one (also Internet Archive copy):
There is currently a forged letter going round on the internet. In the alleged historical writing of 1907, the University of Bern rejects Albert Einstein’s habilitation. The university archivist Niklaus Bütikofer explains which errors the forger had made and tells the true story by means of the original documents.
Some of the errors that are pointed out are laid out in a nice bullet point list:
- At the indicated date, the Faculty of Philosophy and History and the Faculty of Philosophy and Natural Sciences had not been separated yet.
- There has never been a dean – and not even a lecturer – named Wilhelm Heinrich at the University of Bern.
- The language of correspondence between Einstein who was a German native naturalised in 1901, and the university, if true, would have been in German and not in English. Bütikofer suspects that the forger obviously cannot speak any German.
- The stamp next to the signature of «Dean Heinrich» has no relationship whatsoever with the University of Bern. It seems to rather show the coat of arms of Hungary.
- The Sidlerstrasse listed in the forged letter has only existed since 1931. Before this, it was called Sternwartsstrasse, and today’s post code system also did not exist at the time.
The article also discusses signs that the letter was not written on a typewriter at all, but rather that a typewriter font was used on an image of an already folded paper. The use of Einstein 25c USA postage stamps is another sign that the letter would not be genuine, even if affixing those to the letter itself had made sense.
In summary, the alleged letter is referred to as a joke of a bored physics student by University of Bern archivist Niklaus Bütikofer.
Einstein was born in Germany, speaking German. The Universität Bern is a university in Switzerland. The local language in Bern at the time was Swiss German, the written language in use was German.
Even today, the Universität Bern communicates with students in German, unless requested otherwise. The default language of the university website is German.
Edit: Apart from the language mistake, pretty much everything else - other than the font - is wrong too, as has been pointed out in comments and in the other answer. The original handwritten response can be translated as:
Dear Mr Doctor! The Faculty of Philosophy took into consideration your matter with regard to the habilitation in today’s sitting. As the habilitation thesis required by section 2 of the regulation on habilitation does not exist as part of your evidences, the faculty did not accept your application at this time. As soon as you submit such, we will be able to process your application. – With high esteem, The Dean G. Tobler"
Well, just have a look at the stamp: An Einstein stamp, which would mean a Phd candidate already has his own stamp. Obviously the guy who made this letter did it as a joke and was not trying to fool people.
Yes, it is obviously a very badly made fake (bolder than brass).
As already mentioned above the University of Bern would have answered in German and not in English. - Before World War I more scientific articles were published in German Language than in English and French together.