Cyclic codes are an important subclass of linear codes as they have efficient encoding and decoding algorithms and they are widely used in many areas including communication and data storage. The weight distribution is an interesting research subject of cyclic codes. How to construct cyclic codes with only a few weights is an important research problem in the coding theory. To solve this problem, the augmented codes of a class of cyclic codes are investigated based on quadratic Gauss sums and some basic properties of characters over finite fields. The weight distribution and parameters of the augmented code are presented. It turns out that the augmented codes are projective cyclic codes with only a few weights. Compared with the original cyclic code, the augmented codes have higher transmission efficiency. The duals of the augmented codes are optimal or nearly optimal according to the sphere-packing bound. As a byproduct, the complete weight distribution of the original cyclic codes is also given.