  A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that
is invariant under the natural action of the infinite symmetric group. We give
an explicit algorithm to find Gr\"obner bases for symmetric ideals in the
infinite dimensional polynomial ring $R$. This allows for symbolic computation
in a new class of rings. In particular, we solve the ideal membership problem
for symmetric ideals of $R$.
