Conversational Swarm Intelligence (CSI) is a new technology that enables human groups of potentially any size to hold real-time deliberative conversations online. Modeled on the dynamics of biological swarms, CSI aims to optimize group insights and amplify group intelligence. It uses Large Language Models (LLMs) in a novel framework to structure large-scale conversations, combining the benefits of small-group deliberative reasoning and large-group collective intelligence. In this study, a group of 241 real-time participants were asked to estimate the number of gumballs in a jar by looking at a photo. In one test case, individual participants entered their estimation in a standard survey. In another test case, participants converged on groupwise estimates collaboratively using a prototype CSI text-chat platform called Thinkscape. The results show that when using CSI, the group of 241 participants estimated within 12% of the correct answer, which was significantly more accurate (p<0.001) than the average individual (mean error of 55%) and the survey-based Wisdom of Crowd (error of 25%). The group using CSI was also more accurate than an estimate generated by GPT 4 (error of 42%). This suggests that CSI is a viable method for enabling large, networked groups to hold coherent real-time deliberative conversations that amplify collective intelligence. Because this technology is scalable, it could provide a possible pathway towards building a general-purpose Collective Superintelligence (CSi).