In order to port the performance of trained artificial neural networks (ANNs) to spiking neural networks (SNNs), which can be implemented in neuromorphic hardware with a drastically reduced energy consumption, an efficient ANN to SNN conversion is needed. Previous conversion schemes focused on the representation of the analog output of a rectified linear (ReLU) gate in the ANN by the firing rate of a spiking neuron. But this is not possible for other commonly used ANN gates, and it reduces the throughput even for ReLU gates. We introduce a new conversion method where a gate in the ANN, which can basically be of any type, is emulated by a small circuit of spiking neurons, with At Most One Spike (AMOS) per neuron. We show that this AMOS conversion improves the accuracy of SNNs for ImageNet from 74.60% to 80.97%, thereby bringing it within reach of the best available ANN accuracy (85.0%). The Top5 accuracy of SNNs is raised to 95.82%, getting even closer to the best Top5 performance of 97.2% for ANNs. In addition, AMOS conversion improves latency and throughput of spike-based image classification by several orders of magnitude. Hence these results suggest that SNNs provide a viable direction for developing highly energy efficient hardware for AI that combines high performance with versatility of applications.