The quote is one aspect of a big picture. Call the designer's clients to find how the designer worked with them, and what their satisfaction level was and still is (this is a potential long-term business relationship).

What is the designer's system for producing a website, deadlines and approval process? Check the designer's portfolio - do the websites "look" like they are designed professionally? If these aspects are not reviewed by you prior to your website being produced, you will be disappointed with your final product.