Ambient Internet of Things networks use low-cost, low-power backscatter tags in various industry applications. By exploiting those tags, we introduce the integrated sensing and backscatter communication (ISABC) system, featuring multiple backscatter tags, a user (reader), and a full-duplex base station (BS) that integrates sensing and (backscatter) communications. The BS undertakes dual roles of detecting backscatter tags and communicating with the user, leveraging the same temporal and frequency resources. The tag-reflected BS signals offer data to the user and enable the BS to sense the environment simultaneously. We derive both user and tag communication rates and the sensing rate of the BS. We jointly optimize the transmit/received beamformers and tag reflection coefficients to minimize the total BS power. To solve this problem, we employ the alternating optimization technique. We offer a closed-form solution for the received beamformers while utilizing semi-definite relaxation and slack-optimization for transmit beamformers and power reflection coefficients, respectively. For example, with ten transmit/reception antennas at the BS, ISABC delivers a 75% sum communication and sensing rates gain over a traditional backscatter while requiring a 3.4% increase in transmit power. Furthermore, ISABC with active tags only requires a 0.24% increase in transmit power over conventional integrated sensing and communication.