Only one person dies


#1

The king can ask 1000 people to have a sip each, only the person who drank the poison will die after 4 weeks. Question does not limit the number of prisoners in any way.


#2
Read the question carefully 
"Furthermore, it takes one month to have an effect"

#3

1000 people drink one sip each out of 1000 bottles respectively.
1 month after this, 1/1000 people will die and you will know which bottle had poison.


#4

I think it should be 1 and also there is a way which will include 1 prisoner into the experiment.
Hear me out.
We have 5 weeks remaining for the party i.e. 5*7 = 35 days and poison needs 30 or 29 or 31 days to take effect. So, we will simply take 1 prisoner and give him a sip of each bottle in interval of 10 mins. Note the timings. Afterwards note the time of his death also and by subtracting 1 month from death time we will get time at which he drank poisonous wine. Thus we got the poisonous bottle with 1 prisoner only.

Correct me if I am wrong.