Assuming that the data are distributed as Poisson conditional on the population size, then you can obtain confidence intervals for the Poisson rate using the GENMOD procedure as follows:
proc genmod data=mydata;
model birth_count = / dist=poisson offset=log_PopSize;
estimate "log(rate)" intercept 1;
run;
where log_PopSize is the (natural) log of the population size and birth_count should be self explanatory. The above code would provide the rate for a population of size 1. You probably don't want to specify the rate for a population of size 1. In order to specify the rate for a population of size K, compute log_PopSize as
log_PopSize = log(PopSize / K);
Of course, the variable log_PopSize needs to be constructed before executing the GENMOD procedure. You might also want to examine whether a negative binomial distribution specification provides a better fit than the Poisson.