The mass of the W boson is one of the most important parameters in the Standard Model. A precise measurement of the W boson mass, together with the top quark mass, can constrain the mass of the undiscovered Higgs boson within the Standard Model framework or give a hint for physics beyond the Standard Model. A proposal for a measurement of the W boson mass via its two leptonic decay channels $W\rightarrow e\nu_e$ and $W\rightarrow \mu\nu_{\mu}$ is presented. The data sample to be used in this analysis corresponds to 2 fb$^{-1}$ of $p\bar p$ collision data taken with the CDF detector at a center-of-mass energy $\sqrt{s}=1.96$ TeV. This proposal begins with a brief introduction to the Standard Model electroweak theory and motivation for precise $W$ mass measurement, followed by a discussion of the CDF detector and the detection of $W$ decay products. A measurement strategy and a brief summary are presented at the end of this proposal.