The authors consider the Dirichlet boundary value problem for the degenerate elliptic equation $$ \roman{div}(A\nabla u)=0 \tag1 $$ in the upper half-space $\Bbb{R}^{n+1}_+=\{(x,t)\colon x\in\Bbb{R}^n, t\in(0,\infty)\}$, for $n\ge2$. Here, $A$ is a square matrix of order $n+1$, with entries depending on the variable $x$ only, and it is not assumed to be symmetric. The following degenerate ellipticity condition is assumed: there exist constants $0<\lambda\le\Lambda<\infty$ and an $A_2$-weight $\mu$ on $\Bbb{R}^n$ such that the inequalities $$ |\langle A(x)\xi,\zeta\rangle|\le\Lambda\mu(x)|\xi||\zeta|\quad\text{and}\quad |\langle A(x)\xi,\xi\rangle|\ge\lambda\mu(x)|\xi|^2 $$ hold for all $\xi,\zeta\in\Bbb{R}^{n+1}$ and almost all $x\in\Bbb{R}^n$. \par A suitable notion of weak solutions to (1) is given, in the local weighted Sobolev space $W^{1,2}_{\mu,\roman{loc}}$ (also introduced in the paper). In order to define the convergence of a solution $u$ to boundary data, the authors introduce the non-tangential maximal function $N_*u$ defined by $$ (N_*u)(x)=\sup_{(y,t)\in\Gamma(x)}|u(y,t)| $$ for all $x\in\Bbb{R}^n$, where $\Gamma(x)=\{(y,t)\in\Bbb{R}^{n+1}\colon|x-y|< t\}$. \par Then, for $p\in(1,\infty)$ and $f\in L_{\mu}^p(\Bbb{R}^n)$, the Dirichlet problem for equation (1) is defined as follows: $$ \cases \roman{div}(A\nabla u)=0\quad\text{in }\Bbb{R}^{n+1}_+,\\ N_*u\in L^p_{\mu}(\Bbb{R}^n),\\ \lim_{t\to0} u(\cdot,t)=f, \endcases \tag2 $$ where the last limit is required to converge in the $L^p_{\mu}(\Bbb{R}^n)$-norm and in a non-tangential sense. \par In the main result of the paper, it is proved that for every $A$ satisfying the above conditions, there exists $p\in(0,\infty)$ such that problem (2) is solvable for every ${f\in L_{\mu}^p(\Bbb{R}^n)}$. Moreover, an associated notion of elliptic measure $\omega$ is introduced, and the authors prove that this measure belongs to the class $A_\infty(\mu)$. \par As a crucial tool for the proof of the existence result for the solution of the Dirichlet problem, the authors also establish Carleson-type estimates for solutions of equation~(1).