Simulations with General Circulation Models have indicated that global warming will be enhanced at high latitudes. Wetter conditions in these regions are quite plausible with global warming due to warmer sea surface temperatures, melting of sea ice and a greater moisture holding capacity of the atmosphere. Recent observations show a marked increase in precipitation in the High Arctic regions during the past decades. To make reliable predictions of the response of High Arctic glaciers to a warmer and wetter climate, and hence, their contribution to sea-level rise, an adaptive-grid finite-volume glacier model coupled with a surface mass balance model has been developed to investigate the interactions of the glaciers with climate change induced by global warming. The adaptive-grid finite-volume glacier model is an implicit one-dimensional dynamic flowline model. The discretized implicit finite-volume equations are solved by an iterative predictor-corrector method. The grid adapts as the terminus moves in response to changes in surface mass balance. Only the terminus grid point and the penultimate grid point are adapted as the terminus position changes in order to minimise computation. The surface mass balance model consists of two sub-models, an accumulation model and an ablation model. The accumulation model simply assumes that solid precipitation is the main process of accumulation and occurs when the air temperature is below a critical value. Ablation at the glacier surface is determined by the ablation model from the surface energy balance. The coupled model is applied to Austre Broggerbreen, Svalbard, Norway, and White Glacier, N.W.T., Canada. The two glaciers are in regions of different climate and represent two types of climates in the High Arctic. The two glaciers are subjected to 5 possible greenhouse scenarios with various precipitation increases. Since the glaciers are not in balance with the current climate, the results suggested that even with no further warming, High Arctic ice masses would continue to retreat under the current climatic conditions and would contribute to the global sea-level rise of a rate of ~0.21 mm/yr. With the predicted warming, the retreat would be accelerated and would contribute to the global sea-level rise of upto ~0.27 mm/yr. Even a significant increase in precipitation (+60% per century) with the warming would not halt the High Arctic ice masses retreat and would only reduce the contribution to the global sea-level rise to a rate of ~0.19 mm/yr. It is fair to assume uncertainty limits of the predicted results to be ±10%.. This implies the predicted sea-level rises for the global warming scenarios have an uncertainty band of ±0.027 mm yr$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. .