Abstract: It is previously known that the one dimensional mortar finite element projection is stable in the $L^2$ norm, provided that the ratio of any two neighboring mesh intervals is uniformly bounded, but with the constant in the bound depending on the maximum value of that ratio. In this paper, we show that this projection is stable in the $L^2$ norm, independently of the properties of the nonmortar mesh. The 1D trace of the mortar space considered here is a piecewise polynomial space of arbitrary degree; therefore, our result can be used for both the $h$ and the $hp$ version of the mortar finite element.