That sounds like a negative result DOES mean you don't have it, but a positive result DOESNT mean you do.
Edit: I guess this is just misleading and you actually can get false negatives.
They're correct to say it has a 100% detection rate (assuming it does, but we have no reason to suspect otherwise), but I think this is quite misleading - it's detecting HIV-specific antibodies, not the virus itself.
if (test_taken == true) { test_result = true; }
No negatives means no false negatives. ;)
But seriously, this is the classic trade-off between high sensitivity (no false negatives) and high specificity (essentially no false positives). Usually, the lower you set your threshold for detection the more likely you are to tell some people they have the disease when they actually don't. Where you set that bar depends on what you're trying to accomplish and forces you consider the harms associated with telling people they have disease when they don't or missing disease in someone who does.