Created a UV sphere and lopped off half, so a hemisphere remains.
At present there are by default two panes in the UV Editing tab screen--UV Editor and 3D Viewport, side by side.
After Edit Mode!UV!Unwrap in the right pane (in the 3D Viewport), the UV Editor (left pane) changes from a square grid and a cursor in the lower left, into a round grid that can accept a round map--where I've thus done so:
Meanwhile the 3D Viewport (right pane) contains the dome perspective, but it's still gray.
I want to map the .png in the left pane onto this dome in the right pane. But how?