The answer
Search for your chosen name and check the exact result ending in .at. A search result and a basket do not reserve a name.
Ordering requires a customer account with a verified email address. Carefully check the spelling, holder details and total price. Only successful registration secures the domain name.
What to do
Search for a name and add an available .at domain to your basket.
Sign in or create an account and verify your email.
Check contact details and the order summary, read the notices and complete payment.
Check the order status, registration confirmation and your account.