We report observations of a new quasistatic nonequilibrium phenomenon in th
e density of electronic states (DOS) of a superconductor. Tunneling measure
ments of the DOS of ultrathin Al films, at the spin paramagnetically limite
d parallel magnetic field transition, reveal a strongly hysteretic DOS spec
trum. We show that the hysteresis can be characterized as a quasistatic sta
re memory effect in which the state of a film (normal or superconducting) i
s determined by its state prior to entering the hysteretic region. We also
show that state memory can lead to the onset of superconductivity with incr
easing temperature, i.e., reentrance.