We present deep optical observations of the gravitational lens system CLASS B0218+357 from which we derive an estimate for the Hubble Constant ( H _ { 0 } ) . Extensive radio observations using the VLA , MERLIN , the VLBA and VLBI have reduced the degeneracies between H _ { 0 } and the mass model parameters in this lens to one involving only the position of the radio-quiet lensing galaxy with respect to the lensed images . B0218+357 has an image separation of only 334 mas , so optical observations have , up until now , been unable to resolve the lens galaxy from the bright lensed images . Using the new Advanced Camera for Surveys , installed on the Hubble Space Telescope in 2002 , we have obtained deep optical images of the lens system and surrounding field . These observations have allowed us to determine the separation between the lens galaxy centre and the brightest image , and so estimate H _ { 0 } . We find an optical galaxy position – and hence an H _ { 0 } value – that varies depending on our approach to the spiral arms in B0218+357 . If the most prominent spiral arms are left unmasked , we find H _ { 0 } = 70 \pm 5 km s ^ { -1 } Mpc ^ { -1 } ( 95 % confidence ) . If the spiral arms are masked out we find H _ { 0 } = 61 \pm 7 km s ^ { -1 } Mpc ^ { -1 } ( 95 % confidence ) .