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We present the first comprehensive look at the 0.35–5 μm transmission spectrum of the warm (∼800 K) Neptune HAT-P-11b derived from 13 individual transits observed using the Hubble and Spitzer Space Telescopes. Along with the previously published molecular absorption feature in the 1.1–1.7 μm bandpass, we detect a distinct absorption feature at 1.15 μm and a weak feature at 0.95 μm, indicating the presence of water and/or methane with a combined significance of 4.4σ. We find that this planet's nearly flat optical transmission spectrum and attenuated near-infrared molecular absorption features are best matched by models incorporating a high-altitude cloud layer. Atmospheric retrievals using the combined 0.35–1.7 μm Hubble Space Telescope (HST) transmission spectrum yield strong constraints on atmospheric cloud-top pressure and metallicity, but we are unable to match the relatively shallow Spitzer transit depths without underpredicting the strength of the near-infrared molecular absorption bands. HAT-P-11b's HST transmission spectrum is well matched by predictions from our microphysical cloud models. Both forward models and retrievals indicate that HAT-P-11b most likely has a relatively low atmospheric metallicity (<4.6 Z⊙ and <86 Z⊙ at the 2σ and 3σ levels respectively), in contrast to the expected trend based on the solar system planets. Our work also demonstrates that the wide wavelength coverage provided by the addition of the HST STIS data is critical for making these inferences.