You can use stack from the base package. But, you need first to coerce your matrix to a data.frame and to reorder the columns once the data is stacked.
mat <- as.data.frame(mat)
res <- data.frame(time= mat$time,stack(mat,select=-time))
res[,c(3,1,2)]
ind time values
1 C_0 0.0 0.1
2 C_0 0.5 0.2
3 C_0 1.0 0.3
4 C_1 0.0 0.3
5 C_1 0.5 0.4
6 C_1 1.0 0.5
Note that stack is generally more efficient than the reshape2 package.