A Magneto-Optical Trap (MOT) for $^{87}Rb$ atoms near the surface of a dielectric coated mirror at the top of a small $20{ imes}25{ imes}40;mm^3$ cell has been observed. Two beams of $3.3;mW/cm^2$ were used for optical cooling and an anti-Helmholtz magnetic field with a spatial gradient of 9.1 G/cm was used for magnetic trapping. The thickness of the mirror coated on a cover glass was less than $100{mu}m$. The mirror covered the top of a cell and the atom-chip was located outside the vacuum in order to exploit the long life time of the mirror and easy operation of the chip. The trapping position was found 5 mm beneath the mirror surface. The number of trapped atoms was roughly $3{ imes}10^7$ atoms and the temperature was approximately a few tens mK. In this paper, we describe the construction of the mirror-MOT in detail.