Security against eavesdropping is one of the key concerns in the design of any communication system. Many common considerations of the security of a wireless communication channel rely on comparing the signal level measured by Bob (the intended receiver) to that accessible to Eve (an eavesdropper). Frameworks such as Wyner's wiretap model ensure the security of a link, in an average sense, when Bob's signal-to-noise ratio exceeds Eve's. Unfortunately, because these guarantees rely on statistical assumptions about noise, Eve can still occasionally succeed in decoding information. The goal of achieving exactly zero probability of intercept over an engineered region of the broadcast sector, which we term absolute security, remains elusive. Here, we describe the first architecture for a wireless link which provides absolute security. Our approach relies on the inherent properties of broadband and high-gain antennas, and is therefore ideally suited for implementation in millimeter-wave and terahertz wireless systems, where such antennas will generally be employed. We exploit spatial minima of the antenna pattern at different frequencies, the union of which defines a wide region where Eve is guaranteed to fail regardless of her computational capabilities, and regardless of the noise in the channels. Unlike conventional zero-forcing beam forming methods, we show that, for realistic assumptions about the antenna configuration and power budget, this absolute security guarantee can be achieved over most possible eavesdropper locations. Since we use relatively simple frequency-multiplexed coding, together with the underlying physics of a diffracting aperture, this idea is broadly applicable in many contexts.