A novel design of dual-band circularly polarized patch antenna is proposed for millimeter-wave applications of the future generations of mobile communication handsets. The antenna structure is composed of a primary circular patch and four parasitic printed elements. The circular patch has two notches on its perimeter. The primary circular patch is designed to operate at 38 GHz as its first-order resonance with perfect impedance matching over the frequency band (37- 38.5 GHz). Four parasitic elements with Y-shaped slots are capacitively coupled to the primary patch to get it operational at an additional higher frequency with perfect impedance matching over a wideband (48.2-50.1 GHz). Defects in the ground plane are made as two square slots. The notches on the circumference of the circular patch and the square slots in the ground plane are aligned to a diagonal that makes 45o with the axis of symmetry of the feeding line. The width of each slot and the diagonal distance between them are the design parameters that can be set to produce circular polarization with satisfactory axial ratio (AR). The outline of the composite planar antenna structure has a square shape so as not to degrade the circular polarization. Also, to satisfy the impedance matching without disturbing the circular polarization, the primary patch is fed through a tapered microstrip line instead of using a microstrip line with inset feed. The achieved gain, AR, and radiation efficiency are 6.6 dBi, 0.6 dB, and 92%, respectively, at 38 GHz and are 6.7 dBi, 1.6 dB, and 75%, respectively, at 50 GHz. The antenna is printed on a substrate of thickness 0.25 mm and the outer dimensions of the planar composite patch structure are 2:9 mm x 2:9 mm. The antenna bandwidth to provide good circular polarization with perfect impedance matching at the same time are 800 MHz (37-37.8 GHz) and 200MHz (49.9-50.1 GHz) for the lower and higher frequency bands, respectively. The proposed antenna is fabricated, and its performance assessed by simulation is validated by comparison with the results of microwave measurements. Both the simulation and measurement results show good performance of the proposed antenna over the lower and higher frequency bands of operation.