  The bright low-mass X-ray binary (LMXB) GX 13+1 is one of the most peculiar
Galactic binary systems. A periodicity of 24.27 d with a formal statistical
error of 0.03 d was observed in its power spectrum density obtained with RXTE
All Sky Monitor (ASM) data spanning 14 years. Starting from a recent study,
indicating GX 13+1 as a possible dipping source candidate, we systematically
searched for periodic dips in the X-ray light curves of GX 13+1 from 1996 up to
2013 using RXTE/ASM, and MAXI data to determine for the first time the X-ray
orbital ephemeris of GX 13+1. We searched for a periodic signal in the ASM and
MAXI light curves, finding a common periodicity of 24.53 d. We folded the 1.3-5
keV and 5-12.1 keV ASM light curves and the 2-4 and 4-10 keV MAXI light curves
at the period of 24.53 d finding a periodic dip. To refine the value of the
period we used the timing technique dividing the ASM light curve in eight
intervals and the MAXI light curve in two intervals, obtaining four and two dip
arrival times from the ASM and MAXI light curves, respectively. We improved the
X-ray position of GX 13+1 using a recent Chandra observation. The new X-ray
position is discrepant by $\sim 7\arcsec$ from the previous one, while it is
compatible with the infrared and radio counterpart positions. We detected an
X-ray dip, that is totally covered by the Chandra observation, in the light
curve of GX 13+1 and showed, a-posteriori, that it is a periodic dip. We
obtained seven dip arrival times from ASM, MAXI, and Chandra light curves. We
calculated the delays of the detected dip arrival times with respect to the
expected times for a 24.52 d periodicity. Fitting the delays with a linear
function we find that the orbital period and the epoch of reference of GX 13+1
are 24.5274(2) days and 50,086.79(3) MJD, respectively.(Abridged)
