Examine the branch closely for kinking or for damaged bark.

I will check that tomorrow, Logician.

Probably root rot.

https://pnwhandbooks.org/plantdisease/host-disease/rhododendron-phytophthora-root-rot

Same site can also be searched using genus name "rhododendron" for other causes of branch die-back.

The dry grass behind the rhododendron tells me the chance for root rot is small. I tend to think it's a damaged branch.

Phytophthora etc. occur on above ground stems of rhododendrons here as well as roots. Situation does not have to be wet for these to be present. Nor does dying back have to be dispersed throughout the shrub for it to be an infestation. Nothing specific can be determined in this instance without close examination.

Home late tonight; will go out tomorrow and check on your suggestions. I very much appreciate your thoughtful responses. Love this rhody - it has been our leafy barrier to the busy street for 25 years.

Okay - the branch is too high to photograph its bark, but it has a very "old" appearance compared with the rest - it's not vigorous looking. I realize that's pretty subjective. FWIW, the rhody was transplanted into a berm of new soil in about 1990 and has seemed to thrive until two years ago, when the first die-off of this kind took place. The dry grass someone pointed out above isn't grass but vinca at the foot of the berm - in summer 2015, we didn't water and the vinca got very dry and is still coming back. If it is root weevil underground or phytophthora above ground, and I must replant on that spot to make a green hedge, would it be unwise to plant another rhody, and instead plant a bush or bushes that are resistant to root weevil and phytophthora? Thanks, all, L

I don't think root weevil will cause just one branch to die. Highly unlikely.

Mike

Need to find out specifically what is going on before moving to preparing a response. If you are in the US you should seek guidance from your local USDA Cooperative Extension office.