In this paper, a wideband dual-circularly-polarized (dual-CP) antenna array which consists of dual-CP magneto-electric dipole (ME-dipole) elements at $K$-band is designed. The dual-CP ME-dipole contains two input ports for exciting LHCP and RHCP radiated waves separately. The use of two sequential rotation (SR) feeding networks significantly improves the impedance match and the purity of circular polarization (CP) of the ME-dipole array. The resulting array's overall working frequency band is 18 GHz-24 GHz. Within this band, the reflection coefficients of two ports are below −10 dB, the port isolation is better than 10 dB, the side lobe level (SLL) of the radiation pattern is below −10 dB, and axial ratio (AR) is below 3 dB. Meanwhile, the main beams of RHCP wave and LHCP wave have maximum gain values of 16.2 dBi and 15.1 dBi, respectively, while the gain fluctuations are less than 3 dB within the operational band. The antenna array designed in this paper can be well applied in various kinds of communication systems, for example, radar systems and satellite communications.