This paper considers the problem of mapping virtual links to physical network paths, referred to as Virtual Link Embedding (VLE), under the condition that bandwidth demands of virtual links are uncertain. To realize virtual links with predictable performance, the mapping is required to guarantee a bound on the congestion probability of the physical paths that embed the virtual links. To this end, we consider a general uncertainty model in which bandwidth demands of virtual links are expressed by random variables for which only the mean and variance (or a range) are known. We formulate the VLE problem as a nonlinear optimization program and design an algorithm called Equal Partition VLE (epVLE) to solve the problem by employing an approximate formulation that results in a second-order cone program (SOCP) that can be solved efficiently even for large networks. We then provide simulation results as well as model-driven and trace-driven experimental results from an SDN testbed to show the utility and efficiency of the epVLE algorithm in various network scenarios. We apply epVLE to commonly studied small networks as well as randomly generated large networks. Our results show that epVLE is able to satisfy the required link congestion constraint, and that it produces results that are very close to those obtained from the exact optimization model.