The Bunimovich stadium \(S\) is a planar domain given by the union of a rectangle \(R=\{ (x,y)\mid x\in[-\alpha,\alpha], y\in [-\beta,\beta] \}\) with two semicircular regions \(W\) centered at \((\pm\alpha\),0) with radius \(\beta\) lying outside \(R\). Dirichlet eigenfunctions \(u_\lambda\) satisfying the equation \((\Delta-\lambda^2)u_\lambda=0\) on \(S\) are considered. The lower bound \(c\lambda^{-2}\) on the \(L^2\) mass of \(u_\lambda\) in \(W\) is obtained under the assumption of \(L^2\)-normalizing \(u_\lambda\). It is shown that the \(L^2\) norm of \(u_\lambda\) may be controlled by the integral of \(w| \partial_Nu| ^2\) along \(\partial S\cap W\), where \(w\) is a smooth factor on \(W\) vanishing at \(R\cap W\).