suppose you have 1 realisation of x and your estimator is c(x)

For your estimator to be unbiased you require:

where the summation is taken over all possible values of x (1,2,3,4,5,6...)

Now, consider the following function:

c(x)=1 if x=1

c(x)=0 otherwise

The expected value of our function is then

as required

So your estimator is

where c(x) was defined above.

In reality, this estimator is no practical use. But it is an unbiased which is all the question asked for. I did assume that your sample is a single realisation from the distribution. if you have a sample with multiple data points, just discard all but one of them and the proceedure still works.