We demonstrate sorting of rare cancer cells from blood using a thin ribbon monolayer of cells within a credit-card sized, microfluidic laboratory-on-a-card ("lab card") structure. This enables higher cell throughput per minute thereby speeding up cell interrogation. In this approach, multiple cells are viewed and sorted, not individually, but as a whole cell row or section of the ribbon at a time. Gated selection of only the cell rows containing a tagged rare cell provides enrichment of the rare cell relative to background blood cells. We also designed the cell injector for laminar flow antibody labeling within 20s. The approach combines rapid laminar flow cell labeling with monolayer cell sorting thereby enabling rare cell target detection at sensitivity levels 1000 to 10,000 times that of existing flow cytometers. Using this method, total cell labeling and data acquisition time on card may be reduced to a few minutes compared to 30-60 min for standard flow methods.