We measure the evolution of the stellar mass function ( SMF ) from z = 0 - 1 using multi-wavelength imaging and spectroscopic redshifts from the PRism MUlti-object Survey ( PRIMUS ) and the Sloan Digital Sky Survey ( SDSS ) . From PRIMUS we construct an i < 23 flux-limited sample of \sim 40 , 000 galaxies at z = 0.2 - 1.0 over five fields totaling \approx 5.5 deg ^ { 2 } , and from the SDSS we select \sim 170 , 000 galaxies at z = 0.01 - 0.2 that we analyze consistently with respect to PRIMUS to minimize systematic errors in our evolutionary measurements . We find that the SMF of all galaxies evolves relatively little since z = 1 , although we do find evidence for mass assembly downsizing ; we measure a \approx 30 \% increase in the number density of \sim 10 ^ { 10 } \mathcal { M } _ { \sun } galaxies since z \approx 0.6 , and a \lesssim 10 \% change in the number density of all \gtrsim 10 ^ { 11 } \mathcal { M } _ { \sun } galaxies since z \approx 1 . Dividing the sample into star-forming and quiescent using an evolving cut in specific star-formation rate , we find that the number density of \sim 10 ^ { 10 } \mathcal { M } _ { \sun } star-forming galaxies stays relatively constant since z \approx 0.6 , whereas the space-density of \gtrsim 10 ^ { 11 } \mathcal { M } _ { \sun } star-forming galaxies decreases by \approx 50 \% between z \approx 1 and z \approx 0 . Meanwhile , the number density of \sim 10 ^ { 10 } \mathcal { M } _ { \sun } quiescent galaxies increases steeply towards low redshift , by a factor of \sim 2 - 3 since z \approx 0.6 , while the number of massive quiescent galaxies remains approximately constant since z \approx 1 . These results suggest that the rate at which star-forming galaxies are quenched increases with decreasing stellar mass , but that the bulk of the stellar mass buildup within the quiescent population occurs around \sim 10 ^ { 10.8 } \mathcal { M } _ { \sun } . In addition , we conclude that mergers do not appear to be a dominant channel for the stellar mass buildup of galaxies at z < 1 , even among massive ( \gtrsim 10 ^ { 11 } \mathcal { M } _ { \sun } ) quiescent galaxies .