The design of your site can occur at several different moments in the process, but it actually doesn't really matter that much when it does occur.
Except for very basic layout considerations, like whether you will have a right hand information bar, like we use on this site, the process of designing the site is really more creating a so called 'skin' for the content to fit into.
This is not in any way to lower the importance of your site's design, since the average user will probably judge your site by two main factors: 1, does it look reasonably professional?, and 2, can they find what they were looking for on your site easily.
The first condition is a result of well done design, the second is a result of good information architecture and well accessible content.
Generally, we will be working either with one of our designers, or with your designer. If we are using our designer, we will have an agreement on a certain number of drafts of designs for you to approve. In general it's a good idea for you to use a designer whose work you already know and like, since it can cut the costs and time down substantially.
If you are using your own designer, we will need to work with them to make sure that the design they come up with is technically workable (designers tend to want to use elements, like Flash, that have too many negative consequences to consider as major components of your site).
Once you have approved a basic design template, we turn the graphic design to a web page coder, who will recreate the design using code, and some images. The top considerations are always page functionality, versatility, and search engine friendliness, while keeping as close to the design as possible.
We tend to value the designer highly, since they are the ones that create the first impression your pages will create, and there is only one first impression.