Low-energy magnetic excitations in the spin-1/2 chain compound (C$_6$H$_9$N$_2$)CuCl$_3$ [known as (6MAP)CuCl$_3$] are probed by means of tunable-frequency electron spin resonance. Two modes with asymmetric (with respect to the $h u=gmu_B B$ line) frequency-field dependences are resolved, illuminating the striking incompatibility with a simple uniform $S=frac{1}{2}$ Heisenberg chain model. The unusual ESR spectrum is explained in terms of the recently developed theory for spin-1/2 chains, suggesting the important role of next-nearest-neighbor interactions in this compound. Our conclusion is supported by model calculations for the magnetic susceptibility of (6MAP)CuCl$_3$, revealing a good qualitative agreement with experiment.
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