I'm a computer programming student with a heart for web design. Is there a big demand for web designers out in the real world?

There is definitely a big demand for web designers. Like Monty said, there are many facets to what web design can mean: UX (user experience), IA (information architecture), UI (user interface), illustration, typography, etc. There is also a large and still growing demand for developers who have design sensibilities who can help build web interfaces and inform UX, UI, etc. Organizations interested in the design-leaning-developer is often not going to want the developer to define color, typography, create images or illustrations per say, but rather focus on the layout, interactive elements, etc.

As a programmer if you stick to front end development your work will remain closely tied with the interface and design of the web application. That will mean writing javascript, CSS and html. If you are studying Java, Python or any other back end languages in school that is still definitely valuable even if you move into front end work after school. The principles you learn will help support you in following good practices in javascript, or any other language.

Where you work can also determine how involved you can be in the design side of things. Some organizations offer more flexibility around the boundaries of your role. In design focused shops, or any place where the designer and developer work closely with each other, there is generally a lot of opportunity to collaborate on design thinking and impact the aesthetics and interface of what you're building. This is something you can look for and ask about in interviews.

There is a demand for web design but you need to break down the elements and specialize. There is the visual design side of things with layout and color. If you lean heavily in that direction you can focus on graphics design and user experience. There is the web 2.0/Javascript/dynamic web pages area. Here you need to learn a little programming but it will let you make the interactive web pages that companies use.

Web design is a moving field and you will need to always keep you skills up to date. The more things you can do the more valuable you will be.

Demand is definitely there with web designers, but perhaps you can also consider mobile app development as the current trend seems to be heading towards mobile. And every company will eventually have to go mobile in the near future - perhaps that will be the time you'll enter the work force.

When you do want to decide which specialization you want to study in, ask yourself this question:

Are you more frustrated when users of your website are having hard time navigating or understanding your content
or
are you more frustrated when your website/content is broken?

If you say yes to the first option, your interest might be on UX design, visual design, UI design.. etc
If you say yes to the second option, your interest might be on programming and coding.. etc.

You can definitely be equally frustrated when both cases happen, and that's great! Then you can work your ways into becoming of those rare hybrids who are both well versed in visual design and programming skills. These talented people are hard to come across, so if you do become one in the future, I assure you will be in high demand.