A simple method is presented to obtain a high-gain dual-band electromagnetic band gap (EBG) resonator antenna. The antenna is based on a one-dimensional EBG structure, made out of two low-cost unprinted dielectric slabs. The EBG structure is implemented as the antenna superstrate, which has been designed to provide a locally-inverted, positive reflection phase gradient with high reflectivity, in two pre-determined frequency bands. A composite dual-band antenna has been designed and tested with a stacked patch feed. Experimental results confirm the dual-band performance of the prototype antenna. Measured peak gains of 14.5 dBi and 15.1 dBi, and 3-dB gain bandwidth of 4.5% and 4.6%, are achieved at 10.6 GHz and 13.2 GHz, respectively. Measured 10-dB return-loss bandwidths are 6.4% and 3.9% in lower and upper bands, respectively. Potential enhancements of antenna radiation characteristics are studied using small 2$\,\times\,$2 patch array feeds. It was found that such feeds can lead to lower side lobes, higher peak gains and larger gain bandwidths.