  We prove that for each $k\ge0$, the probability that a root vertex in a
random planar graph has degree $k$ tends to a computable constant $d_k$, so
that the expected number of vertices of degree $k$ is asymptotically $d_k n$,
and moreover that $\sum_k d_k =1$.
  The proof uses the tools developed by Gimenez and Noy in their solution to
the problem of the asymptotic enumeration of planar graphs, and is based on a
detailed analysis of the generating functions involved in counting planar
graphs. However, in order to keep track of the degree of the root, new
technical difficulties arise. We obtain explicit, although quite involved
expressions, for the coefficients in the singular expansions of the generating
functions of interest, which allow us to use transfer theorems in order to get
an explicit expression for the probability generating function $p(w)=\sum_k d_k
w^k$. From this we can compute the $d_k$ to any degree of accuracy, and derive
the asymptotic estimate $d_k \sim c\cdot k^{-1/2} q^k$ for large values of $k$,
where $q \approx 0.67$ is a constant defined analytically.
