Axions/axion-like particles (ALPs) are a well-motivated extension of the Standard Model and are generic within String Theory. The X-ray transparency of the intracluster medium (ICM) in galaxy clusters is a powerful probe of light ALPs (with mass <10(-11)eV); as X-ray photons from an embedded or background source propagate through the magnetized ICM, they may undergo energy-dependent quantum mechanical conversion into ALPs (and vice versa), imprinting distortions on the X-ray spectrum. We present Chandra data for the active galactic nucleus NGC 1275 at the center of the Perseus cluster. Employing a 490 ks High Energy Transmission Gratings exposure, we obtain a high-quality 1-9 keV spectrum free from photon pileup and ICM contamination. Apart from iron-band features, the spectrum is described by a power-law continuum, with any spectral distortions at the <3% level. We compute photon survival probabilities as a function of ALP mass m(a) and ALP-photon coupling constant g(alpha gamma) for an ensemble of ICM magnetic field models, and then use the NGC 1275 spectrum to constrain the (m(a), g(alpha gamma))-plane. Marginalizing over magnetic field realizations, the 99.7% credible region limits the ALP-photon coupling to G(alpha gamma) < 6 - 8 x 10(-13) GeV-1 (depending upon magnetic field model) for masses m(a) < 1 x 10(-12) eV. These are the most stringent limit to date on g(alpha gamma) for these light ALPs, and have already reached the sensitivity limits of next-generation helioscopes and light-shining-through-wall experiments. We highlight the potential of these studies with the next-generation X-ray observatories Athena and Lynx, but note the critical importance of advances in relative calibration of these future X-ray spectrometers.