A new silver(I) inorganic polymer, formulated as {[Ag4(H2ebidc) (NH3)4](H2O)2}n (H2ebidc = 2,2'-(ethane-1,2-diyl)bis(1H-imidazole-4,5-dicarboxylic acid), was synthesized and characterized by X-ray crystallography, elemental analysis and IR spectroscopy. The crystal analysis results showed that the four silver(I) ions were linked by the H2ebidc4− ligand to form a tetranuclear cluster subunit, and then a one-dimensional ring-to-ring chain was constructed through the Ag(I)-Ag(I) bonds between the cluster subunits. Electrochemical sensing performance of the Ag(I) inorganic polymer modified carbon paste electrode (CPE-Ag) for H2O2 was evaluated by chronoamperometry in 0.2 M phosphate buffer saline (PBS, pH = 6) at −0.2 V. The electrochemical H2O2 sensor CPE-Ag was found to lead to a linear response from 0.5 μM to 4.0 mM with a detection limit of 0.32 μM. Moreover, the sensor also revealed good anti-interference ability and stability. The above studies proved that Ag(I) inorganic polymers can be used as effective components of electrode materials for the electrochemical recognition of H2O2.