A Chandra image of the central X-ray source RX J0007.0+7303 in the supernova remnant CTA 1 reveals a point source, a compact nebula, and a bent jet, all of which are characteristic of energetic, rotation-powered pulsars. Using the MDM 2.4m telescope we obtain upper limits in the optical at the position of the point source, (J2000) 00h07m01.56s, +73o03'08.1", determined to an accuracy of 0.1", of B>25.4, V>24.8, R>25.1, and I>23.7; these correspond to an X-ray-to-optical flux ratio >100. Neither a VLA image at 1425 MHz, nor a deep pulsar search at 820 MHz using the NRAO Green Bank Telescope, reveal a radio pulsar counterpart to an equivalent luminosity limit at 1400 MHz of 0.02 mJy kpc^2, which is equal to the lowest luminosity known for any radio pulsar. The Chandra point source accounts for about 30% of the flux of RX J0007.0+7303, while its compact nebula plus jet comprise about 70%. The X-ray spectrum of the point source is fitted with a power-law plus blackbody model with Gamma = 1.6+/-0.6, kT_{infty} = 0.13+/-0.05 keV, and R_infty = 0.37 km, values typical of a young pulsar. An upper limit of T^{infty}_e < 6.6x10^5 K to the effective temperature of the entire neutron star surface is derived, which is one of the most constraining data points on cooling models. The 0.5-10 keV luminosity of RX J0007.0+7303 is 4x10^{31}(d/1.4 kpc)^2 ergs s^{-1}, but the larger ~18' diameter synchrotron nebula in which it is embedded is two orders of magnitude more luminous. These properties allow us to estimate, albeit crudely, that the spin-down luminosity of the underlying pulsar is in the range 10^{36}-10^{37} ergs s^{-1}, and support the identification of the high-energy gamma-ray source 3EG J0010+7309 as a pulsar even though its spin parameters have not yet been determined. Comment: 22 pages, 6 figures, submitted to The Astrophysical Journal