How can I prevent Visual Studio 2013 from closing my IIS Express app when I end debugging?

Question

Turning off the new "Enable Edit and Continue" feature fixed it for me.

1. Open Options dialog box (Tools | Options)
2. Locate "Debugging\Edit and Continue"
3. Uncheck "Enable Edit and Continue"

Update 1:
You can also turn it off on a per project basis.

1. Open properties for web project
2. Select Web tab
3. Uncheck "Enable Edit and Continue" in the "Debuggers" section

Update 2: Blog article that discusses this feature.

“Enable Edit and Continue” debugging option is now on by default for new web applications in VS2013 preview

Easier solution:

From Debug menu select Start Without Debugging or Ctrl+F5 to run the project, it will keep running your project unless you quit IIS.

I didn't have "Enable Edit and Continue" in my web project's properties (VS2015 Community Update 2), but finally I found a useful comment in this link which mentioned in Rick's answer:

Christian: You don't need to turn the option off for IIS express to keep running. All you need to do is instead of stopping your application, detach all processes. If you go: Tools > Customize > click the commands tab. Then select toolbar radio button and on the drop down next to this select debug. You then want to "Add Command…". On Add Command window select debug and scroll down to and click on "Detach All". Click OK and then just close. You will see a red cross next to your debug menu. When you click this after debugging your application IIS Express will continue to run.

Briefly: