It will offer current accounts which will be consistent with the principles of Islamic law or sharia.
This prohibits banks from dealing with businesses that are considered sinful, such as alcohol, weapons, gambling and pornography. Usury is also prohibited under sharia law so in principle banks cannot charge fees or interest for money lending.
Banks generally get around this by lending out their capital but in ways where interest and fees are not explicit. So if a client wants to buy a house the bank buys it for them, and sells it back to them at a profit but allows them to pay in instalments.
There will be three different BAWAG sharia accounts available which will not pay or charge any interest, but customers will pay fixed charges of €4.90, €11.90 or €34.90 a month for the account, debit card and overdraft facility.
A storm of criticism on the bank's Facebook page on Thursday prompted it to post a statement clarifying that sharia account customers would not be given preferential treatment and that the new accounts would not be financed by conventional account models.
It also asked people to refrain from making extremist or political statements, insults, threats and personal attacks.
Critical comments included “is this a proactive step in the impending Islamization of Europe?" and one user called it "probably one of the saddest days in the history of banking".
BAWAG has said its target group is the almost 600,000 Muslims living in Austria, mainly from the Turkish and Bosnian communities. Islam is Austria's second largest recognised religion, and the bank said it is important "to respond to the different needs of sections of society”.