Yep, agreed. dBm/Hz.
Note that the displayed RBW usually does not exactly equal the true noise bandwidth. If the RBW says "1KHz", it might actually be 1.1 KHz when it is accurately measured. This is usually caused by the fact that you can not use brick wall filters.
Why this is confusing is that usually you can change the RBW and the signal power does not seem to change on the spectrum analyzer face. This happens if you are looking at a single tone, where all the power is concentrated at one frequency. It matters very little if you look at that tone in a 1 KHz bandwidth or a 1 MHz bandwidth, because each filter will let in the same tone, which contains 99.999% of the power.
However if you were looking at a broadband noise source, you would see the power level change 30 dB in going from 1 KHz bandwidth to 1 MHz bandwidth.