We present high-resolution (0.″16) 870 μm Atacama Large Millimeter/submillimeter Array (ALMA) imaging of 16 luminous ({L}{IR}∼ 4× {10}12 {L}ȯ ) submillimeter galaxies (SMGs) from the ALESS survey of the Extended Chandra Deep Field South. This dust imaging traces the dust-obscured star formation in these z∼ 2.5 galaxies on ∼1.3 kpc scales. The emission has a median effective radius of R e = 0.″24 ± 0.″02, corresponding to a typical physical size of {R}e= 1.8 ± 0.2 kpc. We derive a median Sérsic index of n = 0.9 ± 0.2, implying that the dust emission is remarkably disk-like at the current resolution and sensitivity. We use different weighting schemes with the visibilities to search for clumps on 0.″12 (∼1.0 kpc) scales, but we find no significant evidence for clumping in the majority of cases. Indeed, we demonstrate using simulations that the observed morphologies are generally consistent with smooth exponential disks, suggesting that caution should be exercised when identifying candidate clumps in even moderate signal-to-noise ratio interferometric data. We compare our maps to comparable-resolution Hubble Space Telescope {H}160-band images, finding that the stellar morphologies appear significantly more extended and disturbed, and suggesting that major mergers may be responsible for driving the formation of the compact dust disks we observe. The stark contrast between the obscured and unobscured morphologies may also have implications for SED fitting routines that assume the dust is co-located with the optical/near-IR continuum emission. Finally, we discuss the potential of the current bursts of star formation to transform the observed galaxy sizes and light profiles, showing that the z∼ 0 descendants of these SMGs are expected to have stellar masses, effective radii, and gas surface densities consistent with the most compact massive ({M}* ∼ 1–2 × 1011 {M}ȯ ) early-type galaxies observed locally.