Shoot a Silhouette
What if the shot you really want is between you and the sun? Go ahead and take the shot, but consider an artistic variation designed to make the most of this kind of situation: the silhouette. For this photo, for example, I spied a bird that would not wait for me to walk around the tree for better lighting.
Position yourself so that the subject is between you and the bright background. Now turn off the flash and lock your camera's exposure on the background. Usually you can do this by pressing the shutter down halfway, to set the exposure on the bright sky.
If your camera doesn't have an exposure lock feature, you can use exposure compensation instead. Take the picture normally, but with the camera's exposure compensation (usually abbreviated as EV) set to -2 or -3. That should underexpose the photo, rendering the subject a deep black in the process.