  We consider first-passage percolation (FPP) on the triangular lattice with
vertex weights $(t_v)$ whose common distribution function $F$ satisfies
$F(0)=1/2$. This is known as the critical case of FPP because large (critical)
zero-weight clusters allow travel between distant points in time which is
sublinear in the distance. Denoting by $T(0,\partial B(n))$ the first-passage
time from $0$ to $\{x : \|x\|_\infty = n\}$, we show existence of the "time
constant'' and find its exact value to be
  \[
  \lim_{n \to \infty} \frac{T(0,\partial B(n))}{\log n} =
\frac{I}{2\sqrt{3}\pi} \text{ almost surely},
  \]
  where $I = \inf\{x > 0 : F(x) > 1/2\}$ and $F$ is any critical distribution
for $t_v$. This result shows that the time constant is universal and depends
only on the value of $I$. Furthermore, we find the exact value of the limiting
normalized variance, which is also only a function of $I$, under the optimal
moment condition on $F$. The proof method also shows an analogous universality
on other two-dimensional lattices, assuming the time constant exists.
