Graph Convolutional Networks (GCNs) are a popular machine learning model with a wide range of applications in graph analytics, including healthcare, transportation, and finance. However, a GCN trained without privacy protection measures may memorize private interpersonal relationships in the training data through its model parameters. This poses a substantial risk of compromising privacy through link attacks, potentially leading to violations of privacy regulations such as GDPR. To defend against such attacks, a promising approach is to train the GCN with differential privacy (DP), a rigorous framework that provides strong privacy protection by injecting random noise into the training process. However, training a GCN under DP is a highly challenging task. Existing solutions either perturb the graph topology or inject randomness into the graph convolution operations, or overestimate the amount of noise required, resulting in severe distortions of the network's message aggregation and, thus, poor model utility. Motivated by this, we propose GCON, a novel and effective solution for training GCNs with edge differential privacy. GCON leverages the classic idea of perturbing the objective function to satisfy DP and maintains an unaltered graph convolution process. Our rigorous theoretical analysis offers tight, closed-form bounds on the sensitivity of the graph convolution results and quantifies the impact of an edge modification on the trained model parameters. Extensive experiments using multiple benchmark datasets across diverse settings demonstrate the consistent superiority of GCON over existing solutions.