We can't say if this is brown rot from the photos. However, if you send us clear close-ups of the browning leaves or stem cankers, we can be more helpful.  

Here is info on brown rot: https://extension.umd.edu/hgic/brown-rot-ornamental-cherries-trees  Note the last sentence before Management: "Visible presence of the pathogen is easy under wet conditions and appears as powdery tufts of brown-gray spores that are visible on the outside of infected flowers and on infected fruit or twig surfaces."  You may be able to see this yourself.

 But, since the entire tree seems to be wilting, it is most likely that the problem is environmental/cultural.  We notice that the trunk flare is not visible, which means the tree is planted too deeply.  The flare is where the trunk widens before entering the ground.  Trunks should not go into the soil straight like a lollipop.  When the tree (or any plant) is planted too deeply, it slowly dies.  It can happen over a matter of years.  Scroll down to learn more about this issue here: http://extension.umd.edu/sites/extension.umd.edu/files/_docs/programs/hgic/HGIC_Pubs/TreesandShrubs/...

Clay soil, in itself, would not kill your tree.  It is actually fairly fertile and holds moisture well once it gets wet, though it is better when mixed with organic matter. 

ECN

Here is a close up of the leaves. Does this indicate brown rot? Thanks again for your help. 

Unfortunately, once leaves are completely dead/brown, it's almost impossible to say what caused it.  We would need to see one in the "browning" stage, i.e. only part brown, the rest green, to id brown rot.

However!
Since the entire tree is wilting, that does not suggest brown rot.  It suggests a root problem, probably drowning since we've had so much rain.  It's very possible for one area (clay soil) to drain so poorly that water sits in the planting hole and drowns the tree.   When lots of organic amendments are added to the backfill soil, rain gets into the planting hole and then the surrounding clay (which doesn't drain well) won't let the rain out, so the rain just sits there for days--this is called the "bathtub" effect.

ECN 

Thank you. My sense is that it is in fact dying from the bathtub effect. I want to plant another tree, possible a yoshino cherry or weeping cherry in the same place. How can I avoid this when the clay is so bad?

One of the most important things you can do is to make sure you don't plant the new tree too deeply. Dig the planting hole as deep as the height of the root ball (no deeper) and twice as wide as the width of the root ball. If the hole is too deep the weight of the ball will cause it to sink and you will get the "bathtub effect" again.
Read our information about the planting process for additional guidance. https://extension.umd.edu/hgic/trees-and-shrubs/planting-process
Here also is a good video showing how to plant a container grown tree.
https://youtu.be/d2ZTBBJYljQ
Also, avoid putting mulch directly up against the trunk of the tree. Mulch around the base of the tree no more than 3 inches deep and at least 6 inches away from the trunk.

ckc