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We propose an "extended Schmidt law" with explicit dependence of the star formation efficiency (SFE = SFR/Mgas) on the stellar mass surface density (Σstar). This relation has a power-law index of 0.48 ± 0.04 and a lσ observed scatter on the SFE of 0.4 dex, which holds over five orders of magnitude in the stellar density for individual global galaxies, including various types and especially the low-surface-brightness (LSB) galaxies that deviate significantly from the Kennicutt-Schmidt (KS) law. When applying it to regions of a sample of 12 spiral galaxies at sub-kiloparsec resolution, the extended Schmidt law not only holds for LSB regions but also shows significantly smaller scatters both within and across galaxies compared with the KS law. We argue that this new relation points to the role of existing stars in regulating the SFE, thus better encoding the star formation physics. Comparison with physical models of star formation recipes shows that the extended Schmidt law can be reproduced by some models including gas free fall in a stellar-gravitational potential and pressure-supported star formation. By implementing this new law into the analytic model of gas accretion in ACDM, we show that it can reproduce the observed main sequence of star-forming galaxies (a relation between the SFR and stellar mass) from z = 0 up to z = 2. [ABSTRACT FROM AUTHOR]