assuming you are referring to being in the first batch, either is fine. assume it will take about 2 weeks for the DAT to be verified in house, then .5 to 1 week for AADSAS to send it out, once they have them.
The first batch seems to be sent out the last week of June.... so i would feel confident taking the DAT anytime prior to June 10th would still allow you to be in the first batch.
though for all of you who will ask how late is late i spent a fair amount of time looking this up last cycle and for me mid July was my time frame. I really wanted to have my application complete at schools prior to August 1st. so for this time frame take the DAT before July 15th.
For the last group that wants to know the absolute latest time to take the DAT first understand this is relative to the person and their competitiveness. what it comes down to for the majority is a pre-December interview. for those schools that have not filled out their pre-December interview spots you will still need at least a month prior if not two months before December first for schools to have time to wade through thousands of applicant to hopefully find yours. This is why i feel taking it no later then September 15th would be best. though i should note this late is well past my personal comfort zone. and many of you will be back in classes, just adding to the stress.
hope this helps some of you, good luck