We have conducted a Galactic plane survey of methanol masers at 6668 MHz using a seven-beam receiver on the Parkes telescope. Here we present results from the first part, which provides sensitive unbiased coverage of a large region around the Galactic Centre. Details are given for 183 methanol maser sites in the longitude range 345° through the Galactic Centre to 6°. Within 6° of the Galactic Centre, we found 88 maser sites, of which more than half (48) are new discoveries. The masers are confined to a narrow Galactic latitude range, indicative of many sources at the Galactic Centre distance and beyond, and confined to a thin disc population; there is no high-latitude population that might be ascribed to the Galactic bulge. Within 2° of the Galactic Centre the maser velocities all lie between −60 and +77 km s−1 , a range much smaller than the 540 km s−1 range observed in CO. Elsewhere, the maser with highest positive velocity (+107 km s−1) occurs, surprisingly, near longitude 355° and is probably attributable to the Galactic bar. The maser with the most negative velocity (−127 km s−1) is near longitude 346°, within the longitude–velocity locus of the near side of the '3-kpc arm'. It has the most extreme velocity of a clear population of masers associated with the near and far sides of the 3-kpc arm. Closer to the Galactic Centre the maser space density is generally low, except within 0.25 kpc of the Galactic Centre itself, the 'Galactic Centre zone', where it is 50 times higher, which is hinted at by the longitude distribution, and confirmed by the unusual velocities.